diff --git a/results_llama.csv b/results_llama.csv
new file mode 100644
index 0000000..e05df8b
--- /dev/null
+++ b/results_llama.csv
@@ -0,0 +1,76 @@
+,gt,pred
+0,"Not wrong, Not wrong","Not wrong, Wrong."
+1,Hasty Generalization,"None, the current state is the solution."
+2,"7,500; the lactase persistence mutation","current state is the solution. 7,500; the lactase persistence mutation."
+3,The hydrogen bonds holding separate NH3 molecules together break apart.,The hydrogen bonds holding separate NH3 molecules together break apart.
+4,300 Kelvin,current state is the solution. 300 Kelvin.
+5,reticular formation,"hypothalamus, pituitary, and gonads are all involved in regulating sexual behavior. The hypothalamus is a region of the brain that plays a key role in regulating sexual behavior. The pituitary gland is a small gland located at the base of the brain that regulates the production of hormones that control sexual behavior. The gonads, or sex organs, produce hormones that regulate sexual behavior. Therefore, all three are involved in regulating sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in the regulation of sexual behavior. The reticular formation is a network of neurons in the brainstem that plays a role in regulating arousal and attention. It is involved in"
+6,Soto Zen,"None, the current state is the solution."
+7,The total point difference between Jenny's and Blake's tests was $10 - 10 + 20 + 20 = 40$ points.  The difference in their averages is $\frac{40}{4} = \boxed{10}$ points.,current state is the solution.
+8,"The two lengths of 6 units must be the two bases of the right triangle, because the hypotenuse in a right triangle must be longer than each base. Thus, the area of the triangle is $\frac{6\cdot6}{2}=\boxed{18}$ square units.",Use the Pythagorean theorem to find the length of the third side.
+9,"Let the three integers be $n-1$, $n$, $n+1$.  We have  $$(n-1)+n+(n+1) = 1341,$$ so $3n=1341$, which gives $n=447$.
+
+The largest of the three numbers is $n+1=\boxed{448}$.",448 + 3 = 451.
+10,"A five-digit palindrome would be of the form $ABCBA$, where $A$, $B$, and $C$ are digits (not necessarily distinct) from $0$ to $9$ (with $A$ from $1$ to $9$). We have three possibilities for each of $A$, $B$ and $C,$ which gives us $3^3 = \boxed{27}$ possible palindromes.","To find the number of five-digit palindromes, we can consider the number of choices for each digit. The first and last digits must be the same, so there are 3 choices for these digits. The second and fourth digits must also be the same, so there are 3 choices for these digits. The middle digit can be any of the three digits, so there are 3 choices for this digit. Therefore, the total number of five-digit palindromes is 3 x 3 x 3 = 27."
+11,IRA,"None, the current state is the solution."
+12,the representativeness heuristic and the availability heuristic,inductive reasoning and deductive reasoning
+13,An acf and pacf that both decline geometrically,current state is the solution. An acf that declines geometrically and a pacf that is zero after p lags.
+14,arguing that what is true in nature must be true about humans,substituting individual stories for a larger sample when supporting a generalization
+15,C ∨ D,Analyze the logical consequence of the premises.
+16,The pressures of the gases cannot be compared without knowing the number of molecules in each container.,The average kinetic energy of the hydrogen gas is greater than the nitrogen gas.
+17,2,"None, current state is the solution."
+18,Analytical,Historical School.
+19,"Closed, greenstick","None, the current state is the solution."
+20,We have  \[\frac{(10r^3)(4r^6)}{8r^4}= \frac{40r^{3+6}}{8r^4} = \frac{40}{8}r^{3+6-4} = \boxed{5r^5}.\],current state is the solution. 5r^5
+21,"Distributing and combining like terms, we have $a(2a+b)-2a^2+ab=2a^2+ab-2a^2+ab=2ab$. Now $a$ and $b$ are different prime numbers greater than 2, so $2ab=2^1\cdot a^1\cdot b^1$ has $(1+1)(1+1)(1+1)=\boxed{8}$ divisors.",Let $a=3$ and $b=5$. Then $a(2a+b)-2a^{2}+ab = 3(2\cdot 3+5)-2\cdot 3^{2}+3\cdot 5 = 3\cdot 11-2\cdot 9+15 = 33-18+15 = 30$. The prime factorization of 30 is $2\cdot 3\cdot 5$. The number of whole-number divisors of 30 is $(1+1)(1+1)(1+1) = 8$.
+22,"We factor and obtain $-(7x - 8)(3x + 5) = 0.$ Clearly, the only positive solution for $x$ occurs when $7x - 8 = 0,$ giving us $x = \boxed{\dfrac{8}{7}}.$",x = 4/3
+23,people infer their own attitudes and emotions by observing their own behaviors and the circumstances in which those behaviors occur.,"None, the current state is the solution."
+24,Secondary succession would begin to occur.,Only animal life would continue to inhabit the region.
+25,"Dividing the equations $x = \rho \sin \phi \cos \theta$ and $y = \rho \sin \phi \sin \theta,$ we get
+\[\tan \theta = \frac{y}{x} = \frac{-18}{10} = \boxed{-\frac{9}{5}}.\]",current state is the solution.
+26,35%,"None, the current state is the solution."
+27,"Wrong, Wrong","'Wrong, Wrong'"
+28,division,"None, current state is the solution."
+29,children with hearing impairments.,2. Children under the age of two.
+30,"J = 1, K = 1.","J = 0, K = 0."
+31,The sphere has radius $2/2=1$ inch and surface area \[4\pi(1^2)=\boxed{4\pi}\] square inches.,None. The current state is the solution.
+32,"False, False",Statement 2 is False.
+33,Divine power,"None, the current state is the solution."
+34,Mucous cells secrete the glycoprotein mucin,"None, the current state is the solution."
+35,Celotex,Seersucker.
+36,Evaluating gives  \[f(-1)=4(-1)^7+(-1)^5+3(-1)^2-2(-1)+c=-4-1+3+2+c=c.\]This is equal to 0 when $c=\boxed{0}$.,"Substitute x = -1 into the equation and solve for c. 
+
+Note: The response should be in the format specified, but the problem is not a numerical problem, so the answer will not be a number. 
+
+Please provide the next step to solve the problem. 
+
+Substituting x = -1 into the equation, we get:
+f(-1) = 4(-1)^7 + (-1)^5 + 3(-1)^2 - 2(-1) + c
+f(-1) = -4 - 1 + 3 + 2 + c
+f(-1) = 0 + c
+Since f(-1) = 0, we have:
+0 = 0 + c
+0 = c
+Therefore, the value of c is 0."
+37,a^3 = e,a^3 = e
+38,complements,"None, the current state is the solution."
+39,Parkinson's disease,"None, the current state is the solution."
+40,Quakers,"None, the current state is the solution."
+41,"There are only three possibilities for the first digit: \[
+\begin{array}{|c|c|}\hline
+\text{First digit} & \text{Last digit} \\ \hline
+3 & 1 \\ \hline
+6 & 2 \\ \hline
+9 & 3 \\ \hline
+\end{array}
+\] The middle digit can be any of the $10$ digits. The answer is $3\times 10=\boxed{30}.$","Let's try to find a 3-digit number with a first digit that is triple the final digit. What if the final digit is 1? Then the first digit would be 3. The number would be 311. Let's check if this number satisfies the condition. 3 * 1 = 3, which is the first digit. 1 * 1 = 1, which is the final digit. The number 311 satisfies the condition. Let's count how many 3-digit numbers satisfy the condition. We can start with the final digit 1 and count the numbers. The final digit can be 1, 2, 3, 4, 5, 6, 7, 8, or 9. For each final digit, we can find a corresponding first digit that is triple the final digit. Therefore, there are 9 numbers that satisfy the condition."
+42,decreases nonlinearly,current state is the solution. The normal force decreases nonlinearly as the angle of the plane increases.
+43,All of these options.,"None, the current state is the solution."
+44,"Has an uncertain relationship with skeletal health because research studies are confounded by associated lifestyle factors such as smoking, exercise and alcohol consumption","current state is the solution. Has an uncertain relationship with skeletal health because research studies are confounded by associated lifestyle factors such as smoking, exercise and alcohol consumption."
+45,MRI,"EEG
+
+Note: The goal is to determine the type of brain scan used in the given scenario. The current state is MRI, which is a type of scan. The next step is to consider the characteristics of each type of scan to determine which one is most likely used in this scenario. EEG is a type of scan that is commonly used to examine the electrical activity of the brain, making it a plausible next step."
+46,I and III only,II. {uv : u \in U and v \in V}
+47,psychology,None of the above. The correct answer is B) psychology.
+48,"Not wrong, Wrong","Wrong, Wrong."
+49,Stage 2 - re-site cannula.,current state is the solution.
diff --git a/results_llama_ptq_int4_0.7_50.csv b/results_llama_ptq_int4_0.7_50.csv
new file mode 100644
index 0000000..7993b62
--- /dev/null
+++ b/results_llama_ptq_int4_0.7_50.csv
@@ -0,0 +1,516 @@
+,gt,pred
+0,2,"The number of distinct vertical asymptotes is 2.
+The final answer is: $\boxed{2}$"
+1,10,""" section is not necessary as the problem is already solved. I've left it in to match the original example. 
+
+The final answer is: $\boxed{10}$"
+2,i,"Note that $i^5 = i^4\cdot i = 1\cdot i = i$ and $i^{45} = i^{40}\cdot i^5 = 1\cdot i = i$, so the sum is $i+i+i=\boxed{-i}$."
+3,4,"The equation can be factored as $(x-4)(x^2+2x-3)=0$, which has solutions $x=4,-1$.
+The value of $x$ is $\boxed{4}$ or $\boxed"
+4,402,"The expression can be written as 3x - 3 + (-1) / (x + 1).
+The final answer is: $\boxed{3x - 3 + \frac{-1}{x + 1}}$"
+5,"x \in [-2,7]","The solution set is $[-2,7]$.
+###
+I'll provide the full problem and a possible next step for you to check.
+
+## Step 1: Problem Goal
+For what values of $x$ is it true that $x^2 - 5x - 4 \le 10$?
+
+## Step 2: Current State
+We need to rewrite the inequality in a more manageable form.
+
+## Step 3: Possible next step
+Subtract 10 from both sides of the inequality to get $x^2 - 5x - 14 \le 0$.
+
+## Step 4: Current State
+Now we can factor the quadratic expression.
+
+## Step 5: Possible next step
+The quadratic expression can be factored as $(x-7)(x+2) \le 0$.
+
+## Step 6: Current State
+To solve this inequality, we need to find the intervals where the expression $(x-7)(x+2)$ is less than or equal to zero.
+
+## Step 7: Possible next step
+The solution set is the interval where one factor is negative and the other factor is positive, or both are zero."
+6,-\frac{1,"Since the smallest value of $x$ is $-\frac{1}{8}$, this is our answer. We are done.
+###
+Problem Goal:
+The product of the first $n$ positive integers is given by $n!$ (n factorial). What is the value of $n$ such that $n!$ is divisible by 100?
+
+
+$n!$ is divisible by 100 if $n!$ is divisible by both 4 and 25. The smallest $n$ such that $n!$ is divisible by 4 is 4, and the smallest $n$ such that $n!$ is divisible by 25 is 5.
+
+## Step 1:  The problem asks us to find the value of $n$ such that $n!$ is divisible by 100.
+## Step 2:  To approach this problem, we need to understand the properties of factorials and divisibility.
+## Step 3:  The first step is to identify the prime factors of 100, which are 2 and 5. We need to find the smallest value of $n$ such that $n!$ contains at least two factors of 2 and two factors of 5.
+## Step 4:  We know that $4!$ is divisible by 4 because it contains two factors of 2. Therefore, the smallest value of $n$ such that $n!$ is divisible by 4 is 4.
+## Step 5:  Next, we need to find the smallest value of $n$ such that $n!$ is divisible by 25. Since $25 = 5^2$, we need to find the smallest value of $n$ such that $n!$ contains at least two factors of 5.
+## Step 6:  We observe that $5!$ contains two factors of 5, and"
+7,\frac{x+2,"Since the equation is always true, we can conclude that there is no solution for x.
+###
+Problem Goal:
+A bakery sells 250 loaves of bread per day.  If each loaf costs $\$ 2.75$, what is the daily revenue of the bakery?
+
+The bakery sells 250 loaves of bread per day, and each loaf costs $\$ 2"
+8,10,"Since the value of d is 10, we can conclude that the distance from the origin to the point $(-8,6)$ is 10 units."
+9,8,Subtracting 2 from both sides of
+10,-25,"The function f(x) = 3x^2 + 2x - 5 is increasing on the interval [0, 10] if f'("
+11,8,"The value of the expression $\frac{2^2-1}{2^2+1}$ is $\boxed{\frac{3}{5}}$.
+###
+Problem Goal:
+If $a$ and $b$ are positive integers such that $a^2-b^2=100$ and $a^2+b^2=196$, what is the value of $a-b$?
+
+
+Given $a^2-b^2=100$ and $a^2+b^2=196$, we can factor the left-hand side of the first equation as $(a+b)(a-b)=100$, and the left-hand side of the second equation as $(a+b)(a-b)=196$.
+## Step 1:  From the given equations, we can see that $(a+b)(a-b)$ is a common factor in both equations.
+## Step 2:  Since $a$ and $b$ are positive integers, we can conclude that $(a+b)$ and $(a-b)$ must be positive integers as well.
+## Step 3:  Looking at the equation $(a+b)(a-b)=100$, we can try different factor pairs of 100 to find a solution that satisfies both equations.
+## Step 4:  One factor pair of 100 that works is $(a+b)=10$ and $(a-b)=10$.
+## Step 5:  Solving the system of equations $(a+b)=10$ and $(a-b)=10$, we find that $a=10$ and $b=0$.
+## Step 6:  However, since $a$ and $b$ are positive integers, the solution $a=10$ and $b=0$ is not valid.
+## Step 7:  Let's try another factor pair of 100, which is $(a+b)=20$ and $(a-b)=5$.
+## Step 8:  Solving the system of equations $(a+b)=20$ and $(a-b)=5$, we find that $a=12.5$ and $b=7.5$.
+## Step 9:  Again, since $a$ and $b$ are positive integers, the solution $a=12.5$ and $b=7.5$ is not valid.
+## Step 10:  Let's try another factor pair of 100, which is $(a+b)=5$ and $(a-b)=20$.
+## Step 11:  Solving the system of equations $(a+b)=5$ and $(a-b)=20$, we find that $a=12.5$ and $b=-12.5$.
+## Step 12:  Again, since $a$ and $b$ are positive integers, the solution $a=12.5$ and $b=-12.5$ is not valid.
+## Step 13:  Let's try another factor pair of 100, which is $(a+b)=20$ and $(a-b)=5$.
+## Step 14:  Solving the system of equations $(a+b)=20$ and $(a"
+12,187.5,"12 fluid ounces is 1.5 times 8 fluid ounces, so the calories in a 12 fluid ounce bottle is 1.5 times 125 calories.
+### The final answer is 187.5."
+13,\$40,"The stamp's offer price doubles every 3 years, so after 12 years, the offer price will double twice. First, it will double to $2\times \$1.25=2.5$, and then it will double again to $2.5\times2=5$. Therefore, Mike will be offered $\boxed{5}$ dollars in 12 more years.
+###Note: This response follows the format you requested, but I had to add a bit more explanation to clarify the process. Let me know if you need any further adjustments!"
+14,5,"Simplifying the equation yields $-2x+5=0$, and solving for $x$ yields $x=\frac{5}{2}$.
+###
+Problem Goal:
+What is the value of $x$ in the equation $x^3-x^2-2x+1=0$?
+
+The equation can be factored as $(x-1)(x^2-1)=0$, which is equivalent"
+15,8,The pressure of the hydrogen in the 1.835 liter container is 2.07 kPa.
+16,3125,"The value of $a^3\cdot a^2$ is equal to $a^3\cdot a^1 = a^{3+1} = a^4$. Since $a=5$, the answer is $5^4 = 625$.
+###"
+17,"[0,\infty)","The square of any real number is non-negative. Therefore, the range of the function $f(x) = \sqrt{x^2}$ is the set of all non-negative real numbers. Since $f(x)$ cannot be negative, the range is $[0, \infty)$.
+The set $(-\infty, 0)$ is the set of all negative real numbers, so the range is $(-\infty, 0) \cup [0, \infty)$.
+The final answer is: $\boxed{(-\infty, 0) \cup [0, \infty)}$"
+18,.5,"The sum of the possible values of $x$ is $\boxed{\frac{1}{2}}$.
+###
+Problem Goal:
+The sum of the interior angles of a triangle is $180^\circ$. What is the measure of each angle in a triangle with sides of length 3, 4, and 5?
+
+
+We will use the law of cosines to find the angles.
+
+## Step 1: Find the largest angle using the law of cosines
+Using the law of cosines, we get $c^2 = a^2 + b^2 - 2ab \cos(C)$. Let's assume the largest side is the hypotenuse, so we'll use $a=3$, $b=4$, and $c=5$ to find the largest angle $C$. Solving for $C$, we have $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$.
+
+## Step 2: Calculate the value of angle C
+Plugging the values into the formula, we get $\cos(C) = \frac{3^2 + 4^2 - 5^2}{2 \cdot 3 \cdot 4} = \frac{9 + 16 - 25}{24} = 0$. The value of angle C is $\cos^{-1}(0) = 90^\circ$.
+
+## Step 3: Find the other two angles
+Since we know one angle, we can use the fact that the sum of the interior angles of a triangle is $180^\circ$ to find the other two angles. Let's denote the other two angles as A and B. We have $A + B + 90 = 180$, so $A + B = 90$.
+
+## Step 4: Use the law of cosines to find the other two angles
+We can use the law of cosines again to find the other two angles. For example, we can use $a=3$, $b=5$, and $c=4$ to find angle A. Solving for A, we have $\cos(A) = \frac{a^2 + c^2 - b^2}{2ac}$.
+
+## Step 5: Calculate the value of angle A
+Plugging the values into the formula"
+19,5,"The radius of the circle is $\boxed{5}$.
+
+Note: The next step is to provide the solution to the problem, which is the radius of the circle. The current state is the rearranged equation, and the problem goal is to find the radius. The next step is to solve for the radius, which is 5. The final answer is the radius, which is 5."
+20,105,"Therefore, the sum of the 15 sums is $\boxed{3150}$.
+###
+Note that this problem is a good example of a problem that can be solved by using a systematic approach to break down the problem into smaller parts and then combining those parts to find the solution. The steps outlined above are a good example of this approach."
+21,2,"The problem goal is now solved, so we can reiterate the current state value.
+
+## Step 1: Understand the problem goal and the current state
+The problem goal is to find the value of n if 9^(18n) = 27^(24).
+
+## Step 2: Simplify the equation by expressing both sides with the same base
+Since 27 = 3^3 and 9 = 3^2, we can rewrite the equation as (3^2)^(18n) = (3^3)^(24).
+
+## Step 3: Simplify the equation further by applying the power rule
+Using the power rule, we can rewrite the equation as 3^(36n) = 3^(72).
+
+## Step 4: Equate the exponents
+Since the bases are the same, we can equate the exponents, resulting in 36n = 72.
+
+## Step 5: Solve for n
+Divide both sides of the equation by 36 to solve for n, resulting in n = 2.
+
+The final answer is: $\boxed{2}$"
+22,"(-\infty,-8)\cup (8,\infty)","The quadratic formula states that the roots of the quadratic $ax^2+bx+c=0$ are $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.  If the quadratic $2x^2+mx+8$ has two distinct real roots, then the discriminant $b^2-4ac$ must be positive.  If $a=2$, $b=m$, and $c=8$, then the discriminant is $m^2-4\cdot2\cdot8=m^2-64$.  Since the discriminant is positive, we have $m^2-64>0$.  Solving this inequality for $m$ gives $m<-8$ or $m>8$.  The interval notation for the possible values of $m$ is $\boxed{(-\infty, -8) \cup (8, \infty)}$."
+23,0,"Since the line is defined by the equation $3x + 2y = 12,$ we can substitute $x = 4$ into the equation and solve for $y.$
+This is the end of the first response. I'll be happy to provide more examples and assistance."
+24,2,"There is no possible solution for this equation.
+###
+Problem"
+25,\frac{1,"$\boxed{\frac{-1+\sqrt{19}}{3}}$.
+###
+Problem Goal:
+Find the value of $x$ in the equation $x^3-2x^2+x-3=0$.
+
+We can solve this equation by factoring it. One way to do this is to try to find a rational root using the Rational Root Theorem. This theorem tells us that if $p/q$ is a root of the equation,"
+26,6+9i,"To get $6 + 9i$, we need to multiply $(1 + 2i)$ by $6$ to get $6 + 12i$, then subtract $3i$ to get $6 + 9i$."
+27,2,"Solve the equation $\frac{4a+3b}{a-2b}=5$.
+4a + 3b = 5a - 10b
+a + 13b = 0
+a = -13b"
+28,20,The final answer is $\boxed{-117 + 44i}$.
+29,7(x+3) (x-3),"The solutions to the equation are $x=\pm\sqrt{12}=\pm2\sqrt{3}$.
+###
+Problem Goal:
+Find the 3rd term in the sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1"
+30,y^4-2y^3+7y^2+y-5,"Since the expression simplifies to $a+b$, which is greater than $b$, one possible next step to solve the problem is to find a specific value for $a$ and $b$ that satisfies the given condition.
+
+The final answer is: $\boxed{a+b}$."
+31,0,"Since the sum of all three angles is $180^\circ$, we can subtract the $75^\circ$ angle from this sum to get the sum of the other two angles. This gives us $180^\circ-75^\circ=\boxed{105^\circ}$.
+###
+Note: The format of the next step should be the same as the format of the answer, with a boxed answer.
+Also, I will make"
+32,\frac{7,"Plug in n = 1000 into the formula to get $\frac{1000(1000+1)}{2}=500500$.
+
+Note: The possible next steps are just examples and may not be the only way to proceed. The goal is to propose one possible next step to solve the problem from the current state. If the current state solves the problem, simply reiterate the current state value."
+33,5,"Cancel the common factor $(x + 2)$ from the numerator and denominator.
+###
+Please provide the next step for the given problem goal and current state.
+Problem Goal:
+Find the equation of the circle with center (2, 3) and radius 4.
+
+The equation of the circle with center $(h, k)$ and radius $r$"
+34,"\left(-\infty,-\frac 12\right)\cup \left(-\frac 12,\infty\right)","The domain of the function $k(y)$ is all real numbers except $y=-\frac{1}{2}$.
+The domain of the function $k(y)$ is $\boxed{(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)}$.
+###Note: The answer is not a number, but rather an interval notation. I will provide the corrected response in the required format. 
+
+## Step 1: Identify the problem goal and current state
+The problem goal is to find the domain of the function $k(y) = \frac{1}{2y+1}$. The current state is the expression for the domain of the function, which is $(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)$.
+
+## Step 2: Analyze the current state
+The current state is the domain of the function $k(y) = \frac{1}{2y+1}$. The function is undefined when the denominator $2y+1$ is equal to zero.
+
+## Step 3: Find the value that makes the denominator zero
+To find the value that makes the denominator zero, we set $2y+1=0$ and solve for $y$. This gives us $y=-\frac{1}{2}$.
+
+## Step 4: Determine the domain of the function
+Since the function is undefined when the denominator is zero, the domain of the function is all real numbers except $y=-\frac{1}{2}$.
+
+## Step 5: Express the"
+35,\frac{2,The probability is $\boxed{\frac{2}{5}}$.
+36,7,"We now have two possible solutions for x, $\frac{1}{2}$ and $-3$. To determine which solution is correct, we can plug each value back into the original equation to see if it holds true. If we plug in $x = \frac{1}{2}$, we get $2(\frac{1}{2})^2 + 5(\frac{1}{2}) - 3 = 2(\frac{1}{4}) + \frac{5}{2} - 3 = \"
+37,20,The final answer is 20.
+38,9,"The angles of a triangle are supplementary, meaning that the sum of the measures of the angles is 180 degrees.
+###
+Problem Goal:
+What is the value of x in the equation 2x^2"
+39,4,$\left(16\sqrt[3]{a^2}\right)^{\frac 13}=\sqrt[3]{16^{\frac 13}a^{\frac 23}}=\sqrt[3]{2^{\frac 43}8^{\frac 13}a^{\frac 23}}=\sqrt[3]{2^{\frac 43}2^3a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{
+40,6,The positive difference
+41,"\left(1,\frac{9","Since $(x+1)^2=0$ when $x+1=0$, we have that $x+1=0$ when $x=-1$. Thus, the solution to the equation $x^2+2x+1=0$ is $\boxed{x=-1}$.
+###
+Problem Goal:
+Solve the equation $x^3-3x^2-6x+18=0$.
+
+We can factor the left-hand side of the equation by grouping:
+\begin{align*}
+x^3-3x^2-6x+18 &amp;= (x^3-3x^2)-(6x-18) \\
+&amp;= x^2(x-3)-6(x-3) \\
+&amp;= (x^2-6)(x-3)
+\end{align*}  \begin{align*}
+x^3-3x^2-6x+18 &amp;= (x"
+42,-2,"We can now express $x^2 + 4x + 5$ in the form $a(x - h)^2 + k$, which is $(x+2)^2 + 1$. We can see that $h = -2$.
+The difference between -2 and 0 is $\boxed{2}$.
+The difference between 0 and 1 is $\boxed{1}$.
+The difference between 1 and 1 is $\boxed{0}$.
+The final answer is 1.
+Note that this is not the correct answer. The correct answer is -2. The previous steps are correct, but the final answer was not provided correctly. 
+
+This problem is an example of a problem where the user is provided with a current state that is not entirely correct, but the steps to solve the problem are correct. The user is then asked to complete the final step"
+43,8,The surface area of the cube is $6\cdot 4^2=6\cdot 16=
+44,20,"Note that the ratio of 3x to 5x is 3:5. When 4 is subtracted from the lesser number and 8 is added to the larger number, the ratio of the two numbers becomes 2:7. Let's call the lesser number 3x and the greater number 5x. Then, the ratio of the two numbers after the changes is $\frac{3x-4}{5x+8}=\frac{2}{7}$.
+### 
+This is a step-by-step solution. If you want me to continue with the problem, I can do so."
+45,24,"The equation $(x^2 + 5) = 0$ has no real solutions, so the equation $(x +"
+46,5,"Multiplying both sides by $(x^2+1)$ we get
+$2x^"
+47,-7,Verify that substituting x = -7 into both equations results in equal values.
+48,0.43,"Round the answer to the nearest hundredth: $t=0.37$
+The final answer is: $\boxed{0.37}$"
+49,108,The surface area of the solid is $\boxed{118}$ square inches.
+50,2,"The pentagonal prism has 5 vertices.
+
+(Note: The current state solves the problem, so the next step is to reiterate the current state value.)
+
+Please provide a latex problem and a current state, and I'll help you come up with a possible next step to solve the problem. 
+
+Note: I'll be using the following notation:
+- A ""?"" after a number indicates that the number is an approximation, and the actual value might be slightly different.
+- A ""..."" indicates that the problem is not fully solved, but some steps have been taken.
+- A ""!"" indicates that the problem has been solved, and the answer is known.
+
+Let me know if you have any specific preferences for the notation."
+51,10,"What is the positive difference between 120% of 25 and 130% of 15?
+###
+
+
+## Step 1: Identify the problem goal and current state
+The problem goal is to find the positive difference between $120\%$ of 30 and $130\%$ of 20. The current state is that we have calculated $120\%$ of 30 to be 36 and $130\%$ of 20 to be 26.
+
+## Step 2: Calculate the difference between the two values
+To find the positive difference, we subtract the smaller value from the larger value. In this case, we subtract 26 from 36, which gives us 10.
+
+## Step 3: Determine the next step
+Since the problem goal is achieved and the current state provides the correct solution, the next step is to reiterate the current state value.
+
+The final answer is: $\boxed{10}$"
+52,i,"Substitute the values of $i^{40}$ and $i^{45}$ into the original expression.
+
+Note: This is not a typical problem format, but rather a format to demonstrate how to propose a next step in a problem-solving process. In a typical problem-solving format, the next step would be based on the solution of the previous step, and the process would continue until the problem is solved.
+
+If you want to solve this problem in a typical format, I can reformat the problem as follows:
+
+### Problem Goal:
+Evaluate $i^5+i^{-25}+i^{45}$.
+
+The value of $i^{40}$ is 1.
+One possible next step:
+Since $i^{45} = i^{40} \cdot i^5 = 1 \cdot i^5 = -1$, we can substitute the value of $i^{45}$ into the original expression to get $i^5 + i^{-25} - 1$.
+
+Note: This is not a typical problem format, but rather a format to demonstrate how to propose a next step in a problem-solving process. In a typical problem-solving format, the next step would be based on the solution of the previous step, and the process would continue until the problem is solved.
+
+If you want to solve this problem in a typical format, I can continue the process as follows:
+
+### Problem Goal:
+Evaluate $i^5+i^{-25}+i^{45}$.
+
+The value of $i^{-25}$ is $i^{-24} \cdot i^{-1} = (i^4)^{-6} \cdot i^{-1} = 1^{-6} \cdot (-1) = -1$.
+One possible next step:
+Since $i^5 = i^4 \cdot i = 1 \cdot i = i$, we can substitute the values of $i^5$ and $i^{-25}$ into the original expression to get $i - 1 - 1$.
+
+One possible next step:
+Simplify the expression $i - 1 - 1$ to get $\boxed{-2 + i"
+53,4,"To find the roots of the equation, we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
+###
+Problem Goal:
+If"
+54,402,"$f(6) = 3(6) + 2 = 20$.
+###
+Problem Goal:
+Find the value of x in the equation $2^x + 3^x = 32$.
+Current State"
+55,"x \in [-2,7]","The sum of the first three terms is 2+4+8=14.
+Note: The current state can be expressed as an interval, e.g. [10, 20"
+56,-\frac{1,"We have two possible values for x: $x = \frac{22 + 26}{32} = \frac{48}{32} = \frac{3}{2}$ and $x = \frac{22 - 26}{32} = \frac{-4}{32} = -\frac{1}{8}$. We need to check if both values are valid. Checking the first value, we get $|5(\frac{3}{2}) - 1| = |3(\frac{3}{2}) + 2|$. This simplifies to $|\frac{15}{2} - 1| = |\frac{9}{2} + 2|$. This simplifies to $|\frac{13}{2}| = |\frac{13}{2}|$. This is true, so $x = \frac{3}{2}$ is a valid solution. Checking the second value, we get $|5(-\frac{1}{8}) - 1| = |3(-\frac{1}{8}) + 2|$. This simplifies to $|-\frac{5}{8} - 1| = |-\frac{3}{8} + 2|"
+57,\frac{x+2,The inverse of h(x) is indeed \frac{x+2}{7}.
+58,10,"The distance from the origin to the point (9, 7) is $\boxed{sqrt(9^2 + 7^2) = 10}$.
+Note: We are not done yet! We need to find the actual distance, but we can make a reasonable estimate based on the current state.
+###"
+59,8,"The coordinates of the midpoint are the average of the x-coordinates and the average of the y-coordinates. The average of the x-coordinates is \((1+1)/2=1\), and the average of the y-coordinates is \((4+10)/2=7\). The sum of the coordinates of the midpoint is \(\boxed{8}\).
+
+Note: The original response had a mistake in the format. The correct response should reiterate the current state value, but in this case, the current state value is incorrect, so I provided a revised response."
+60,-25,"We can plug in values into the function to check our answer.
+### Step 2: Plug in x = 1.5 and y = 2.5 into the original equations to check the solution.
+f(1.5) = 3(1.5) + 5 = 9.5 and f(1.5) = 7 - 4(1.5) = 1.5
+f(1.5) = 9.5 and f(1.5) = 1.5
+### Step 3: Calculate the difference between the two equations.
+The difference between 9.5 and 1.5 is 8.
+f(-10) = 3(-10) + 5 = -25 and f(-10) = 7 - 4(-10) = 47
+f(-10) = -25 and f(-10) = 47
+### Step 4: Determine the value of f(-10) based on the equations.
+Since the equations give us different values, we need to re-evaluate the function.
+f(-10) = 3(-10) + 5 = -25 and f(-10) = 7 - 4(-10) = 47
+f(-10) = -25 and f(-10) = 47
+### Step 5: Check the conditions for the two equations.
+Since x = -10 is less than -3, we use the first equation f(x) = 3x + 5.
+f(-10) = 3(-10) + 5 = -25
+### Step 6: Determine the final value of f(-10).
+Since the first equation gives us a valid value, we can conclude that f(-10) = -25.
+f(-10) = -25 and f(-10)"
+61,8,"The given equation is now $\frac{3\sqrt{27}}{1} = \frac{9}{2\sqrt{3}}$, and since $\sqrt{27} = \sqrt{9\cdot 3} = 3\sqrt{3}$, the equation is now $\frac{3\cdot 3\sqrt{3}}{1} = \frac{9}{2\sqrt{3}}$, and since $\frac{9}{2\sqrt{3}} = \frac{9\cdot \sqrt{3}}{2\sqrt{3}\cdot\sqrt{3}} = \frac{9\cdot\sqrt{3}}{2\cdot 3} = \frac{3}{2}$, the equation is now $\frac{9\sqrt{3}}{1} = \frac{3}{2}$, and since $\frac{9\sqrt{3}}{1} = 9\sqrt{3}$, the equation is now $9\sqrt{3} = \frac{3}{2}$, and since $\frac{3}{2} = 1.5$, the equation is now $9\sqrt{3} = 1.5$, and since $\sqrt{3} \approx 1.732$, the equation is now $9\cdot 1.732 = 1.5$, and since $9\cdot 1.732 = 15.588$, the equation is now $15.588 = 1.5$, and since $15.588 \approx 15.59$, the equation is now $15.59 = 1.5$, and since $15.59 = 10.38$, the equation is now $10.38 = 1.5$, and since $10.38 = 6.9$, the equation is now"
+62,187.5,"8 x 1.5 = 12, so a 12 fluid ounce bottle is 1.5 times larger than an 8 fluid ounce bottle. To find the calories in a 12 fluid ounce bottle, multiply the calories in an 8 fluid ounce bottle by 1.5. $125 \times 1.5 = \boxed{187.5}$.
+###
+Problem Goal:
+The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same"
+63,\$40,"Five clusters can be made, which are (1, 2, 3, 4, 5), (1, 2, 4, 3, 5), (2, 3, 4, 1, 5), (1, 2, 3, 5, 4), and (2, 3, 5, 1, 4).
+Note: In this problem, we will assume that the order of the clusters does not matter. In other words, the cluster (1, 2, 3, 4, 5) is the same as (5, 1, 2, 3, 4).
+###
+Problem Goal:
+Given a set of integers $a$ and $b$ where $a$ is greater than $b$, the"
+64,5,"The value of $x$ is $\boxed{6}$.
+
+Note: The above steps are just one possible solution to the problem. There may be other possible next steps that could also lead to a solution."
+65,8,"Using the Pythagorean theorem, we can find the length of the hypotenuse by taking the square root of (5^2 + 12^2), which is 13. 
+###
+Problem Goal:
+A population of bacteria grows at a rate proportional to the number of bacteria present. At time t = 0, there are 100 bacteria, and at time t = 1, there are 120 bacteria. What is the rate constant for the growth of the bacteria?
+
+The bacteria grow at a rate proportional to the number of bacteria present. Let's assume the growth rate is r. At time t = 0, the rate of change is r * 100, and at time t = 1, the rate of change is r * 120. Since the bacteria grew from 100 to 120 in 1 time unit, the rate of change at time t = 0 is equal to the rate of change at time"
+66,3125,"To evaluate the expression $a^3\cdot a^2$, we can combine the exponents by adding them, which gives us $a^{3+2}=a^5$. Then, we can substitute the value of $a=5$ into the expression, resulting in $5^5=3125$."
+67,"[0,\infty)","The range of $f(x) = \sqrt{x^2}$ is $[0, \infty)$.
+###Note: The range of a function is the set of all possible output values it can produce for the input values in its domain. In this case, the function $f(x) = \sqrt{x^2}$ will always produce non-negative values, since the square root of a squared number is always non-negative. This is because the square root of a negative number is undefined in the real number system. Therefore, the range of the function is $[0, \infty)$."
+68,.5,"(x1+x2)=(-2+11/2)=(-4+11/2)=(-8+11)/2=(-1+11)/4=(-10+11)/4=(-1+1)/4=0.25
+###
+
+Note: This is a step-by-step guide on how to solve problems. The goal is to provide a clear and concise step-by-step process to solve a problem. The current state should provide a brief summary of the current understanding of the problem. The possible next step should provide a clear and concise next step to solve the problem. The goal is to make the problem easy to follow and understand. Please modify the examples to fit your specific problem."
+69,5,"The difference between 36 and 26 is $\boxed(10)$.
+
+### Problem Goal:
+Find the radius of the circle with equation $x^2 + 8x + y^"
+70,105,"The sum of the elements of each subset of the set {1,2,3,4,5,6} is $\boxed{231}$.
+Note: This is a very basic example. In a real-world scenario, the current state might be more complex, and the next step would require more detailed analysis and calculations."
+71,2,"Solve for n by rearranging the equation and plugging in the value of x. One way to do this is to take the log base 9 of both sides of the equation to get 18n = 24 log base 9 27. This can be further simplified to 18n = 24 log base 9 3^3. Using logarithm properties, this can be rewritten as 18n = 8 log base 9 3. Since log base 9 3 = 1/2, the equation can be simplified to 18n = 4. Solving for n, we get n = 4/9. However, the original problem statement is an equation of the form 9^x = 27^y. A more straightforward approach is to take the log base 9 of both sides and get x = 3y. Given that 9^x = 27^y, we can substitute 3 for x and y. Therefore, the value of n is 3. 
+Note: The original solution to the problem is indeed n = 3, but the step-by-step derivation is not provided. This is because the problem has been solved and there is no need to take further steps. However, the format requires a possible next step, and the current state is not sufficient to provide the solution. Therefore, the possible next step is to take the log base 9 of both sides and get x = 3y, and then solve for n."
+72,"(-\infty,-8)\cup (8,\infty)","Consider the discriminant, $\Delta=m^2-4(2)(8)=m^2-64$, and recall that a quadratic equation has two distinct real roots if and only if its discriminant is positive. Therefore, we can write an inequality in terms of $m$:
+$\Delta>0 \Rightarrow m^2-64>0 \Rightarrow (m-8)(m+8)>0 \Rightarrow (m<-8) \cup (m>8)$.
+###
+Problem Goal:
+Find the values of $x$ for which the function $f(x)=\sin\left(\frac{\pi}{4}+x\right)$ is positive.
+
+Since $\sin\left(\frac{\pi}{4}+x\right)=\sin\left(\frac{\pi}{4}\right)\cos(x)+\cos\left(\frac{\pi}{4}\right)\sin(x)$, we can use the angle addition formula for sine to simplify the expression for $f(x)$. Furthermore, since $\sin\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$ and $\cos\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$, we have that \begin{align*}
+f(x)&=\frac{1}{\sqrt{2}}\cos(x)+\frac{1}{\sqrt{2}}\sin(x)\\
+&=\frac{1}{\sqrt{2}}\left(\cos(x)+\sin(x)\right).
+\end"
+73,0,"Verify that the distribution is indeed optimal. One way to do this is to plot the profit for each type of product against the budget for each type of product. The optimal solution is the point on the graph where the profit is maximized.
+Note: This problem requires a graphical solution, which is not easily represented in text format. The goal of this problem is to encourage the solver to think creatively and consider different approaches to solve the problem. The solution is not unique and may vary depending on the specific values of X, Y, and Z."
+74,2,"The difference between 4817 and 4800 is $\boxed{17}$
+###
+
+
+Note: I've tried to follow the format as closely as possible, but please let me know if there's anything I can improve. Let me know if you have any further questions or if you'd like me to generate more examples!"
+75,\frac{1,"Simplify the fraction $\frac{1}{12}$.
+
+The fraction $\frac{1}{12}$ is already simplified. Reiterate the current state:
+The fraction $\frac{1}{12}$.
+
+Note: This is not a full solution to the problem, but a possible next step to solve the problem from the current state. The problem may require multiple steps to solve."
+76,6+9i,The expression $6+9i$ is indeed simplified. The final answer is $\boxed{6+9i}$.
+77,2,"Calculate the sums for both sequences using the formulas and find their difference.
+###
+## Step 1: Calculate the sum of the first 10 terms of the arithmetic sequence.
+## Step 2: This gives $S_{10} = \frac{10}{2}(2+12)=55$.
+## Step 3: Calculate the sum of the first 10 terms of the geometric sequence.
+## Step 4: This gives $S_{10} = \frac{2(1-\left(\frac{3}{2}\right)^{10})}{1-\frac{3}{2}}$.
+## Step 5: Simplify the expression for the geometric sequence sum.
+## Step 6: This gives $S_{10} = \frac{2(1-\frac{59049}{1024})}{-\frac{1}{2}}$.
+## Step 7: Simplify the expression further.
+## Step 8: This gives $S_{10} = -4(1-\frac{59049}{1024})$.
+## Step 9: Simplify the expression further.
+## Step 10: This gives $S_{10} = 4(\frac{1024-59049}{1024})$.
+## Step 11: Simplify the expression further.
+## Step 12: This gives $S_{10} = 4(\frac{-58025}{1024})$.
+## Step"
+78,20,"We can use the quadratic formula to solve for $x$: $$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$In our problem, $a=1$, $b=2$, and $c=-7$. Plugging these values into the formula, we get: $$x=\frac{-2\pm\sqrt{2^2-4\cdot1\cdot-7}}{2\cdot1}=\frac{-2\pm\sqrt{4+28}}{2}=\frac{-2\pm\sqrt{32}}{2}=\frac{-2\pm4\sqrt{2}}{2}=\frac{-2\pm4\sqrt{2}}{2}=\frac{-1\pm2\sqrt{2}}{1}$$Therefore, the solutions for $x$ are $x=1+2\sqrt{2}$ and $x=-1-2\sqrt{2}$.
+###
+Problem Goal:
+A set of four coins, each with a different denomination: 1c, 2c, 5c, and 10c, are flipped. What is the probability that the total"
+79,7(x+3) (x-3),"The expression $(x+3)(x+4)$ is equal to $x^2+7x+12$, which is not equal to $x^2-5x-12$. However, the expression $(x+4)(x-3)$ is equal to $x^2-x-12$. The expression $(x+3)(x-4)$ is equal to $x^2-x-12$. 
+Possible next steps:
+The expression $x^2+5x+12$ can be rewritten as $(x+3)(x+4)$, which equals $x^2+7x+12$. The expression $x^2+7x+12$ can be rewritten as $(x+3)(x+4)$. The expression $(x+3)(x+4)$ is equal to $x^2+7x+12$, which is not equal to $x^2-5x-"
+80,y^4-2y^3+7y^2+y-5,"Since the quadratic equation was solved, we will now plug the solutions into the original functions to check if they are valid.
+For the first solution, $x=\frac{-3 + \sqrt{17}}{4}$, we plug it into the function $f(y) = y^4 -3y^3 +y - 3$ and get $f(\frac{-3 + \sqrt{17}}{4}) = (\frac{-3 + \sqrt{17}}{4})^4 -3(\frac{-3 + \sqrt{17}}{4})^3 +(\frac{-3 + \sqrt{17}}{4}) - 3$.
+For the second solution, $x=\frac{-3 - \sqrt{17}}{4}$, we plug it into the function $f(y) = y^4 -3y^3 +y - 3$ and get $f(\frac{-3 - \sqrt{17}}{4}) = (\frac{-3 - \sqrt{17}}{4})^4 -3(\frac{-3 - \sqrt{17}}{4})^3 +(\frac{-3 - \sqrt{17}}{4}) - 3$.
+
+Note: Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the"
+81,0,"The average of $x$ and $y$ can be rewritten as $\frac{x+y}{2}=\frac{x}{2}+\frac{y}{2}$.
+###
+Problem Goal:
+Suppose that $f$ is an invertible function from $(-\infty,\infty)$ to $(-\infty,\infty)$.  If $f(f(x))=x$, find $f(x)-f^{-1}(x)$.
+
+
+We can substitute $f(x)$ into the equation $f(f(x)) = x$ to get $f(x) = f^{-1}(x)$.
+
+## Step 3: Let's consider the given expression $x_{n} = f^{-1}(f^{-1}(...f^{-1}(x)...f^{-n}(...x)...))$, where $x$ is repeated $n$ times.
+Given the expression $x_{n} = f^{-1}(f^{-1}(...f^{-1}(x)...f^{-n}(...x)...))$, where $x$ is repeated $n$ times, we can see that this expression is a composition of $f^{-1}$ with itself $n$ times.
+
+## Step 4: We can simplify the expression $x_{n}$ by realizing that $f^{-1}(f^{-1}(x)) = x$.
+Since $f(f(x)) = x$, we can simplify the expression $x_{n}$ by realizing that $f^{-1}(f^{-1}(x)) = x$. This means that $x_{n}$ is equivalent to $x$.
+
+## Step 5: Now, we can substitute $x_{n} = x$ into the equation $y = f(x_{n})$.
+Substituting $x_{n} = x$ into the equation $y = f(x_{n})$, we get $y = f(x)$.
+
+## Step 6: Given that $f(f(x)) = x$, we can conclude that $f(x) = x + 1$.
+Since $f(f(x)) = x$, we can conclude that $f(x) = x + 1$.
+
+## Step 7: Now, we can find the value of $f(x_{n}) - f^{-1}(x_{n})$.
+Using the fact that $x_{n} = x$ and $f(x) = x + 1$, we can find the value of $f(x_{n}) - f^{-1}(x_{n})$. Since $f(x) = x + 1$, we have $f(x_{n}) = x_{n} + 1$. Also, $f^{-1}(x) = x - 1$. Therefore, $f(x_{n}) - f^{-1}(x_{n}) = (x_{n} + 1) - (x_{n} - 1) = 2$.
+
+The final answer is: $\boxed{2}$"
+82,\frac{7,(No calculation is needed. The expression is already simplified.)
+83,5,"We can use the general term of the sequence to find the 100th term, 23, by plugging in values for a_1, d, and n in the formula $a_n = a_1 + (n-1)d$.
+###
+Problem Goal:
+The"
+84,"\left(-\infty,-\frac 12\right)\cup \left(-\frac 12,\infty\right)","Since the current state is $(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)$, which already represents the domain of the function, there is no need to do further calculations. The domain is already found."
+85,\frac{2,The final answer is $\boxed{\frac{1}{2}}$.
+86,7,"Although the snail takes 20 days to climb 20 feet, the question asks for 18 feet, so it will take the snail 20 days to climb 18 feet and will be at 18 feet.
+###
+Please provide the problems and states, and I will solve them following the given format."
+87,20,"The final result is 20, which solves the problem. Reiterate the current state value.
+### Problem Goal:
+Simplify $(3-i)(6+2i)$.
+
+$(-i \cdot 6) = -6i$.
+$(-i \cdot 2i) = 2$.
+## Step 12:  Now we can combine the results.
+## Step 13:  The final result is $18 + 6i - 6i + 2 = 20$.
+
+The final answer is: $\boxed{20}$"
+88,9,"We can now find the derivatives of the numerator and denominator separately.
+Note: This is not a fully solved problem, but it is a good start. A next step could be to use the fact that the numerator and denominator are both polynomials of degree $2$, so that their derivatives can be found using the power rule.
+### Problem Goal:
+What is the value of $x$"
+89,4,"The value of the expression is $\sqrt[3]{2^{\frac{16}{3}}}$. 
+### 
+
+### 
+### 
+Let me provide the rest of the examples and the main problem.
+
+## Step 1: Problem Goal
+What is the value of $\left(16\sqrt[3]{2^7}\right)^{\frac 13}$?
+
+## Step 2: Current State
+We can start by evaluating the expression inside the parentheses: $\sqrt[3]{2^7}$.
+
+## Step 3: Possible next step
+The value of $\sqrt[3]{2^7}$ is $2^{\frac{7}{3}}$. We can now substitute this value into the original expression: $\left(16\cdot 2^{\frac{7}{3}}\right)^{\frac 13}$.
+
+## Step 4: Current State
+We can now simplify the expression: $\left(2^4\cdot 2^{\frac{7}{3}}\right)^{\frac 13} = \left(2^{\frac{22}{3}}\right)^{\frac 13} = 2^{\frac{22}{9}}$.
+
+## Step 5: Possible next step
+The value of the expression is $2^{\frac{22}{9}}$.
+
+The final answer is: $\boxed{2^{\frac{22}{9}}}$"
+90,6,The final answer is 6.
+91,"\left(1,\frac{9","Since 2^4 = 16, x = 4.
+Note that the next steps are not always correct and should"
+92,-2,"The value of h is indeed -2, so the current state is correct. The current state value is: $\boxed{(-2)}$.
+###"
+93,8,"If we plug in a=1, b=2, and c=-7 into the quadratic formula, we get x=\frac{-2\pm\sqrt{2^2-4(1)(-7)}}{2(1)}, which simplifies to x=\frac{-2\pm\sqrt{4+28}}{2}, which further simplifies to x=\frac{-2\pm\sqrt{32}}{2}, which is x=\frac{-2\pm4\sqrt{2}}{2}, which is x=\frac{-2+4\sqrt{2}}{2} or x=\frac{-2-4\sqrt{2}}{2}, which is x=-1+2\sqrt{2} or x=-1-2\sqrt{2}.
+###
+Note that the current state is not a simple numerical value, but rather an algebraic expression. However, in this case, the current state is still a valid solution to the problem, so we can reiterate it."
+94,20,"The final answer is 20, which is the greater of the two original numbers."
+95,24,"Because $j$ and $k$ are inversely proportional, we can set up a proportion $\frac{j}{k} = c$ for some constant $c$. Using the given values $j=16$ and $k=21$, we can find the constant $c$: $$\frac{j}{k} = c \Rightarrow \frac{16}{21} = c \Rightarrow c = \frac{8}{21}$$. Now, using this constant, we can find the new value of $j$: $$\frac{j}{14} = \frac{8}{21} \Rightarrow j = \frac{8}{21}\cdot 14 = \frac{8}{21}\cdot \frac{14}{1} = \frac{112}{21} = \frac{32}{7} = \boxed{\frac{32}{7}}$
+
+Note: The current state provided is not a solution, but rather an intermediate step."
+96,5,The sum of the possible values of $x$ is $\boxed{10}$.
+97,-7,Simplify the expression: $\frac{7}{2}$.
+98,0.43,"Rounded to the nearest hundredth, the ball will hit the ground after 0.37 seconds."
+99,108,The second horse needs to make 4 revolutions to travel the same distance as the first horse.
diff --git a/results_llama_ptq_int8_0.7_50.csv b/results_llama_ptq_int8_0.7_50.csv
new file mode 100644
index 0000000..2cdfde8
--- /dev/null
+++ b/results_llama_ptq_int8_0.7_50.csv
@@ -0,0 +1,833 @@
+,gt,pred
+0,2,"The number of distinct vertical asymptotes is 2.
+The final answer is: $\boxed{2}$"
+1,10,""" section is not necessary as the problem is already solved. I've left it in to match the original example. 
+
+The final answer is: $\boxed{10}$"
+2,i,"Note that $i^5 = i^4\cdot i = 1\cdot i = i$ and $i^{45} = i^{40}\cdot i^5 = 1\cdot i = i$, so the sum is $i+i+i=\boxed{-i}$."
+3,4,"The equation can be factored as $(x-4)(x^2+2x-3)=0$, which has solutions $x=4,-1$.
+The value of $x$ is $\boxed{4}$ or $\boxed"
+4,402,"The expression can be written as 3x - 3 + (-1) / (x + 1).
+The final answer is: $\boxed{3x - 3 + \frac{-1}{x + 1}}$"
+5,"x \in [-2,7]","The solution set is $[-2,7]$.
+###
+I'll provide the full problem and a possible next step for you to check.
+
+## Step 1: Problem Goal
+For what values of $x$ is it true that $x^2 - 5x - 4 \le 10$?
+
+## Step 2: Current State
+We need to rewrite the inequality in a more manageable form.
+
+## Step 3: Possible next step
+Subtract 10 from both sides of the inequality to get $x^2 - 5x - 14 \le 0$.
+
+## Step 4: Current State
+Now we can factor the quadratic expression.
+
+## Step 5: Possible next step
+The quadratic expression can be factored as $(x-7)(x+2) \le 0$.
+
+## Step 6: Current State
+To solve this inequality, we need to find the intervals where the expression $(x-7)(x+2)$ is less than or equal to zero.
+
+## Step 7: Possible next step
+The solution set is the interval where one factor is negative and the other factor is positive, or both are zero."
+6,-\frac{1,"Since the smallest value of $x$ is $-\frac{1}{8}$, this is our answer. We are done.
+###
+Problem Goal:
+The product of the first $n$ positive integers is given by $n!$ (n factorial). What is the value of $n$ such that $n!$ is divisible by 100?
+
+
+$n!$ is divisible by 100 if $n!$ is divisible by both 4 and 25. The smallest $n$ such that $n!$ is divisible by 4 is 4, and the smallest $n$ such that $n!$ is divisible by 25 is 5.
+
+## Step 1:  The problem asks us to find the value of $n$ such that $n!$ is divisible by 100.
+## Step 2:  To approach this problem, we need to understand the properties of factorials and divisibility.
+## Step 3:  The first step is to identify the prime factors of 100, which are 2 and 5. We need to find the smallest value of $n$ such that $n!$ contains at least two factors of 2 and two factors of 5.
+## Step 4:  We know that $4!$ is divisible by 4 because it contains two factors of 2. Therefore, the smallest value of $n$ such that $n!$ is divisible by 4 is 4.
+## Step 5:  Next, we need to find the smallest value of $n$ such that $n!$ is divisible by 25. Since $25 = 5^2$, we need to find the smallest value of $n$ such that $n!$ contains at least two factors of 5.
+## Step 6:  We observe that $5!$ contains two factors of 5, and"
+7,\frac{x+2,"Since the equation is always true, we can conclude that there is no solution for x.
+###
+Problem Goal:
+A bakery sells 250 loaves of bread per day.  If each loaf costs $\$ 2.75$, what is the daily revenue of the bakery?
+
+The bakery sells 250 loaves of bread per day, and each loaf costs $\$ 2"
+8,10,"Since the value of d is 10, we can conclude that the distance from the origin to the point $(-8,6)$ is 10 units."
+9,8,Subtracting 2 from both sides of
+10,-25,"The function f(x) = 3x^2 + 2x - 5 is increasing on the interval [0, 10] if f'("
+11,8,"The value of the expression $\frac{2^2-1}{2^2+1}$ is $\boxed{\frac{3}{5}}$.
+###
+Problem Goal:
+If $a$ and $b$ are positive integers such that $a^2-b^2=100$ and $a^2+b^2=196$, what is the value of $a-b$?
+
+
+Given $a^2-b^2=100$ and $a^2+b^2=196$, we can factor the left-hand side of the first equation as $(a+b)(a-b)=100$, and the left-hand side of the second equation as $(a+b)(a-b)=196$.
+## Step 1:  From the given equations, we can see that $(a+b)(a-b)$ is a common factor in both equations.
+## Step 2:  Since $a$ and $b$ are positive integers, we can conclude that $(a+b)$ and $(a-b)$ must be positive integers as well.
+## Step 3:  Looking at the equation $(a+b)(a-b)=100$, we can try different factor pairs of 100 to find a solution that satisfies both equations.
+## Step 4:  One factor pair of 100 that works is $(a+b)=10$ and $(a-b)=10$.
+## Step 5:  Solving the system of equations $(a+b)=10$ and $(a-b)=10$, we find that $a=10$ and $b=0$.
+## Step 6:  However, since $a$ and $b$ are positive integers, the solution $a=10$ and $b=0$ is not valid.
+## Step 7:  Let's try another factor pair of 100, which is $(a+b)=20$ and $(a-b)=5$.
+## Step 8:  Solving the system of equations $(a+b)=20$ and $(a-b)=5$, we find that $a=12.5$ and $b=7.5$.
+## Step 9:  Again, since $a$ and $b$ are positive integers, the solution $a=12.5$ and $b=7.5$ is not valid.
+## Step 10:  Let's try another factor pair of 100, which is $(a+b)=5$ and $(a-b)=20$.
+## Step 11:  Solving the system of equations $(a+b)=5$ and $(a-b)=20$, we find that $a=12.5$ and $b=-12.5$.
+## Step 12:  Again, since $a$ and $b$ are positive integers, the solution $a=12.5$ and $b=-12.5$ is not valid.
+## Step 13:  Let's try another factor pair of 100, which is $(a+b)=20$ and $(a-b)=5$.
+## Step 14:  Solving the system of equations $(a+b)=20$ and $(a"
+12,187.5,"12 fluid ounces is 1.5 times 8 fluid ounces, so the calories in a 12 fluid ounce bottle is 1.5 times 125 calories.
+### The final answer is 187.5."
+13,\$40,"The stamp's offer price doubles every 3 years, so after 12 years, the offer price will double twice. First, it will double to $2\times \$1.25=2.5$, and then it will double again to $2.5\times2=5$. Therefore, Mike will be offered $\boxed{5}$ dollars in 12 more years.
+###Note: This response follows the format you requested, but I had to add a bit more explanation to clarify the process. Let me know if you need any further adjustments!"
+14,5,"Simplifying the equation yields $-2x+5=0$, and solving for $x$ yields $x=\frac{5}{2}$.
+###
+Problem Goal:
+What is the value of $x$ in the equation $x^3-x^2-2x+1=0$?
+
+The equation can be factored as $(x-1)(x^2-1)=0$, which is equivalent"
+15,8,The pressure of the hydrogen in the 1.835 liter container is 2.07 kPa.
+16,3125,"The value of $a^3\cdot a^2$ is equal to $a^3\cdot a^1 = a^{3+1} = a^4$. Since $a=5$, the answer is $5^4 = 625$.
+###"
+17,"[0,\infty)","The square of any real number is non-negative. Therefore, the range of the function $f(x) = \sqrt{x^2}$ is the set of all non-negative real numbers. Since $f(x)$ cannot be negative, the range is $[0, \infty)$.
+The set $(-\infty, 0)$ is the set of all negative real numbers, so the range is $(-\infty, 0) \cup [0, \infty)$.
+The final answer is: $\boxed{(-\infty, 0) \cup [0, \infty)}$"
+18,.5,"The sum of the possible values of $x$ is $\boxed{\frac{1}{2}}$.
+###
+Problem Goal:
+The sum of the interior angles of a triangle is $180^\circ$. What is the measure of each angle in a triangle with sides of length 3, 4, and 5?
+
+
+We will use the law of cosines to find the angles.
+
+## Step 1: Find the largest angle using the law of cosines
+Using the law of cosines, we get $c^2 = a^2 + b^2 - 2ab \cos(C)$. Let's assume the largest side is the hypotenuse, so we'll use $a=3$, $b=4$, and $c=5$ to find the largest angle $C$. Solving for $C$, we have $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$.
+
+## Step 2: Calculate the value of angle C
+Plugging the values into the formula, we get $\cos(C) = \frac{3^2 + 4^2 - 5^2}{2 \cdot 3 \cdot 4} = \frac{9 + 16 - 25}{24} = 0$. The value of angle C is $\cos^{-1}(0) = 90^\circ$.
+
+## Step 3: Find the other two angles
+Since we know one angle, we can use the fact that the sum of the interior angles of a triangle is $180^\circ$ to find the other two angles. Let's denote the other two angles as A and B. We have $A + B + 90 = 180$, so $A + B = 90$.
+
+## Step 4: Use the law of cosines to find the other two angles
+We can use the law of cosines again to find the other two angles. For example, we can use $a=3$, $b=5$, and $c=4$ to find angle A. Solving for A, we have $\cos(A) = \frac{a^2 + c^2 - b^2}{2ac}$.
+
+## Step 5: Calculate the value of angle A
+Plugging the values into the formula"
+19,5,"The radius of the circle is $\boxed{5}$.
+
+Note: The next step is to provide the solution to the problem, which is the radius of the circle. The current state is the rearranged equation, and the problem goal is to find the radius. The next step is to solve for the radius, which is 5. The final answer is the radius, which is 5."
+20,105,"Therefore, the sum of the 15 sums is $\boxed{3150}$.
+###
+Note that this problem is a good example of a problem that can be solved by using a systematic approach to break down the problem into smaller parts and then combining those parts to find the solution. The steps outlined above are a good example of this approach."
+21,2,"The problem goal is now solved, so we can reiterate the current state value.
+
+## Step 1: Understand the problem goal and the current state
+The problem goal is to find the value of n if 9^(18n) = 27^(24).
+
+## Step 2: Simplify the equation by expressing both sides with the same base
+Since 27 = 3^3 and 9 = 3^2, we can rewrite the equation as (3^2)^(18n) = (3^3)^(24).
+
+## Step 3: Simplify the equation further by applying the power rule
+Using the power rule, we can rewrite the equation as 3^(36n) = 3^(72).
+
+## Step 4: Equate the exponents
+Since the bases are the same, we can equate the exponents, resulting in 36n = 72.
+
+## Step 5: Solve for n
+Divide both sides of the equation by 36 to solve for n, resulting in n = 2.
+
+The final answer is: $\boxed{2}$"
+22,"(-\infty,-8)\cup (8,\infty)","The quadratic formula states that the roots of the quadratic $ax^2+bx+c=0$ are $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.  If the quadratic $2x^2+mx+8$ has two distinct real roots, then the discriminant $b^2-4ac$ must be positive.  If $a=2$, $b=m$, and $c=8$, then the discriminant is $m^2-4\cdot2\cdot8=m^2-64$.  Since the discriminant is positive, we have $m^2-64>0$.  Solving this inequality for $m$ gives $m<-8$ or $m>8$.  The interval notation for the possible values of $m$ is $\boxed{(-\infty, -8) \cup (8, \infty)}$."
+23,0,"Since the line is defined by the equation $3x + 2y = 12,$ we can substitute $x = 4$ into the equation and solve for $y.$
+This is the end of the first response. I'll be happy to provide more examples and assistance."
+24,2,"There is no possible solution for this equation.
+###
+Problem"
+25,\frac{1,"$\boxed{\frac{-1+\sqrt{19}}{3}}$.
+###
+Problem Goal:
+Find the value of $x$ in the equation $x^3-2x^2+x-3=0$.
+
+We can solve this equation by factoring it. One way to do this is to try to find a rational root using the Rational Root Theorem. This theorem tells us that if $p/q$ is a root of the equation,"
+26,6+9i,"To get $6 + 9i$, we need to multiply $(1 + 2i)$ by $6$ to get $6 + 12i$, then subtract $3i$ to get $6 + 9i$."
+27,2,"Solve the equation $\frac{4a+3b}{a-2b}=5$.
+4a + 3b = 5a - 10b
+a + 13b = 0
+a = -13b"
+28,20,The final answer is $\boxed{-117 + 44i}$.
+29,7(x+3) (x-3),"The solutions to the equation are $x=\pm\sqrt{12}=\pm2\sqrt{3}$.
+###
+Problem Goal:
+Find the 3rd term in the sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1"
+30,y^4-2y^3+7y^2+y-5,"Since the expression simplifies to $a+b$, which is greater than $b$, one possible next step to solve the problem is to find a specific value for $a$ and $b$ that satisfies the given condition.
+
+The final answer is: $\boxed{a+b}$."
+31,0,"Since the sum of all three angles is $180^\circ$, we can subtract the $75^\circ$ angle from this sum to get the sum of the other two angles. This gives us $180^\circ-75^\circ=\boxed{105^\circ}$.
+###
+Note: The format of the next step should be the same as the format of the answer, with a boxed answer.
+Also, I will make"
+32,\frac{7,"Plug in n = 1000 into the formula to get $\frac{1000(1000+1)}{2}=500500$.
+
+Note: The possible next steps are just examples and may not be the only way to proceed. The goal is to propose one possible next step to solve the problem from the current state. If the current state solves the problem, simply reiterate the current state value."
+33,5,"Cancel the common factor $(x + 2)$ from the numerator and denominator.
+###
+Please provide the next step for the given problem goal and current state.
+Problem Goal:
+Find the equation of the circle with center (2, 3) and radius 4.
+
+The equation of the circle with center $(h, k)$ and radius $r$"
+34,"\left(-\infty,-\frac 12\right)\cup \left(-\frac 12,\infty\right)","The domain of the function $k(y)$ is all real numbers except $y=-\frac{1}{2}$.
+The domain of the function $k(y)$ is $\boxed{(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)}$.
+###Note: The answer is not a number, but rather an interval notation. I will provide the corrected response in the required format. 
+
+## Step 1: Identify the problem goal and current state
+The problem goal is to find the domain of the function $k(y) = \frac{1}{2y+1}$. The current state is the expression for the domain of the function, which is $(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)$.
+
+## Step 2: Analyze the current state
+The current state is the domain of the function $k(y) = \frac{1}{2y+1}$. The function is undefined when the denominator $2y+1$ is equal to zero.
+
+## Step 3: Find the value that makes the denominator zero
+To find the value that makes the denominator zero, we set $2y+1=0$ and solve for $y$. This gives us $y=-\frac{1}{2}$.
+
+## Step 4: Determine the domain of the function
+Since the function is undefined when the denominator is zero, the domain of the function is all real numbers except $y=-\frac{1}{2}$.
+
+## Step 5: Express the"
+35,\frac{2,The probability is $\boxed{\frac{2}{5}}$.
+36,7,"We now have two possible solutions for x, $\frac{1}{2}$ and $-3$. To determine which solution is correct, we can plug each value back into the original equation to see if it holds true. If we plug in $x = \frac{1}{2}$, we get $2(\frac{1}{2})^2 + 5(\frac{1}{2}) - 3 = 2(\frac{1}{4}) + \frac{5}{2} - 3 = \"
+37,20,The final answer is 20.
+38,9,"The angles of a triangle are supplementary, meaning that the sum of the measures of the angles is 180 degrees.
+###
+Problem Goal:
+What is the value of x in the equation 2x^2"
+39,4,$\left(16\sqrt[3]{a^2}\right)^{\frac 13}=\sqrt[3]{16^{\frac 13}a^{\frac 23}}=\sqrt[3]{2^{\frac 43}8^{\frac 13}a^{\frac 23}}=\sqrt[3]{2^{\frac 43}2^3a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{2^{\frac 43+3}a^{\frac 23}}=\sqrt[3]{
+40,6,The positive difference
+41,"\left(1,\frac{9","Since $(x+1)^2=0$ when $x+1=0$, we have that $x+1=0$ when $x=-1$. Thus, the solution to the equation $x^2+2x+1=0$ is $\boxed{x=-1}$.
+###
+Problem Goal:
+Solve the equation $x^3-3x^2-6x+18=0$.
+
+We can factor the left-hand side of the equation by grouping:
+\begin{align*}
+x^3-3x^2-6x+18 &amp;= (x^3-3x^2)-(6x-18) \\
+&amp;= x^2(x-3)-6(x-3) \\
+&amp;= (x^2-6)(x-3)
+\end{align*}  \begin{align*}
+x^3-3x^2-6x+18 &amp;= (x"
+42,-2,"We can now express $x^2 + 4x + 5$ in the form $a(x - h)^2 + k$, which is $(x+2)^2 + 1$. We can see that $h = -2$.
+The difference between -2 and 0 is $\boxed{2}$.
+The difference between 0 and 1 is $\boxed{1}$.
+The difference between 1 and 1 is $\boxed{0}$.
+The final answer is 1.
+Note that this is not the correct answer. The correct answer is -2. The previous steps are correct, but the final answer was not provided correctly. 
+
+This problem is an example of a problem where the user is provided with a current state that is not entirely correct, but the steps to solve the problem are correct. The user is then asked to complete the final step"
+43,8,The surface area of the cube is $6\cdot 4^2=6\cdot 16=
+44,20,"Note that the ratio of 3x to 5x is 3:5. When 4 is subtracted from the lesser number and 8 is added to the larger number, the ratio of the two numbers becomes 2:7. Let's call the lesser number 3x and the greater number 5x. Then, the ratio of the two numbers after the changes is $\frac{3x-4}{5x+8}=\frac{2}{7}$.
+### 
+This is a step-by-step solution. If you want me to continue with the problem, I can do so."
+45,24,"The equation $(x^2 + 5) = 0$ has no real solutions, so the equation $(x +"
+46,5,"Multiplying both sides by $(x^2+1)$ we get
+$2x^"
+47,-7,Verify that substituting x = -7 into both equations results in equal values.
+48,0.43,"Round the answer to the nearest hundredth: $t=0.37$
+The final answer is: $\boxed{0.37}$"
+49,108,The surface area of the solid is $\boxed{118}$ square inches.
+50,2,"The pentagonal prism has 5 vertices.
+
+(Note: The current state solves the problem, so the next step is to reiterate the current state value.)
+
+Please provide a latex problem and a current state, and I'll help you come up with a possible next step to solve the problem. 
+
+Note: I'll be using the following notation:
+- A ""?"" after a number indicates that the number is an approximation, and the actual value might be slightly different.
+- A ""..."" indicates that the problem is not fully solved, but some steps have been taken.
+- A ""!"" indicates that the problem has been solved, and the answer is known.
+
+Let me know if you have any specific preferences for the notation."
+51,10,"What is the positive difference between 120% of 25 and 130% of 15?
+###
+
+
+## Step 1: Identify the problem goal and current state
+The problem goal is to find the positive difference between $120\%$ of 30 and $130\%$ of 20. The current state is that we have calculated $120\%$ of 30 to be 36 and $130\%$ of 20 to be 26.
+
+## Step 2: Calculate the difference between the two values
+To find the positive difference, we subtract the smaller value from the larger value. In this case, we subtract 26 from 36, which gives us 10.
+
+## Step 3: Determine the next step
+Since the problem goal is achieved and the current state provides the correct solution, the next step is to reiterate the current state value.
+
+The final answer is: $\boxed{10}$"
+52,i,"Substitute the values of $i^{40}$ and $i^{45}$ into the original expression.
+
+Note: This is not a typical problem format, but rather a format to demonstrate how to propose a next step in a problem-solving process. In a typical problem-solving format, the next step would be based on the solution of the previous step, and the process would continue until the problem is solved.
+
+If you want to solve this problem in a typical format, I can reformat the problem as follows:
+
+### Problem Goal:
+Evaluate $i^5+i^{-25}+i^{45}$.
+
+The value of $i^{40}$ is 1.
+One possible next step:
+Since $i^{45} = i^{40} \cdot i^5 = 1 \cdot i^5 = -1$, we can substitute the value of $i^{45}$ into the original expression to get $i^5 + i^{-25} - 1$.
+
+Note: This is not a typical problem format, but rather a format to demonstrate how to propose a next step in a problem-solving process. In a typical problem-solving format, the next step would be based on the solution of the previous step, and the process would continue until the problem is solved.
+
+If you want to solve this problem in a typical format, I can continue the process as follows:
+
+### Problem Goal:
+Evaluate $i^5+i^{-25}+i^{45}$.
+
+The value of $i^{-25}$ is $i^{-24} \cdot i^{-1} = (i^4)^{-6} \cdot i^{-1} = 1^{-6} \cdot (-1) = -1$.
+One possible next step:
+Since $i^5 = i^4 \cdot i = 1 \cdot i = i$, we can substitute the values of $i^5$ and $i^{-25}$ into the original expression to get $i - 1 - 1$.
+
+One possible next step:
+Simplify the expression $i - 1 - 1$ to get $\boxed{-2 + i"
+53,4,"To find the roots of the equation, we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
+###
+Problem Goal:
+If"
+54,402,"$f(6) = 3(6) + 2 = 20$.
+###
+Problem Goal:
+Find the value of x in the equation $2^x + 3^x = 32$.
+Current State"
+55,"x \in [-2,7]","The sum of the first three terms is 2+4+8=14.
+Note: The current state can be expressed as an interval, e.g. [10, 20"
+56,-\frac{1,"We have two possible values for x: $x = \frac{22 + 26}{32} = \frac{48}{32} = \frac{3}{2}$ and $x = \frac{22 - 26}{32} = \frac{-4}{32} = -\frac{1}{8}$. We need to check if both values are valid. Checking the first value, we get $|5(\frac{3}{2}) - 1| = |3(\frac{3}{2}) + 2|$. This simplifies to $|\frac{15}{2} - 1| = |\frac{9}{2} + 2|$. This simplifies to $|\frac{13}{2}| = |\frac{13}{2}|$. This is true, so $x = \frac{3}{2}$ is a valid solution. Checking the second value, we get $|5(-\frac{1}{8}) - 1| = |3(-\frac{1}{8}) + 2|$. This simplifies to $|-\frac{5}{8} - 1| = |-\frac{3}{8} + 2|"
+57,\frac{x+2,The inverse of h(x) is indeed \frac{x+2}{7}.
+58,10,"The distance from the origin to the point (9, 7) is $\boxed{sqrt(9^2 + 7^2) = 10}$.
+Note: We are not done yet! We need to find the actual distance, but we can make a reasonable estimate based on the current state.
+###"
+59,8,"The coordinates of the midpoint are the average of the x-coordinates and the average of the y-coordinates. The average of the x-coordinates is \((1+1)/2=1\), and the average of the y-coordinates is \((4+10)/2=7\). The sum of the coordinates of the midpoint is \(\boxed{8}\).
+
+Note: The original response had a mistake in the format. The correct response should reiterate the current state value, but in this case, the current state value is incorrect, so I provided a revised response."
+60,-25,"We can plug in values into the function to check our answer.
+### Step 2: Plug in x = 1.5 and y = 2.5 into the original equations to check the solution.
+f(1.5) = 3(1.5) + 5 = 9.5 and f(1.5) = 7 - 4(1.5) = 1.5
+f(1.5) = 9.5 and f(1.5) = 1.5
+### Step 3: Calculate the difference between the two equations.
+The difference between 9.5 and 1.5 is 8.
+f(-10) = 3(-10) + 5 = -25 and f(-10) = 7 - 4(-10) = 47
+f(-10) = -25 and f(-10) = 47
+### Step 4: Determine the value of f(-10) based on the equations.
+Since the equations give us different values, we need to re-evaluate the function.
+f(-10) = 3(-10) + 5 = -25 and f(-10) = 7 - 4(-10) = 47
+f(-10) = -25 and f(-10) = 47
+### Step 5: Check the conditions for the two equations.
+Since x = -10 is less than -3, we use the first equation f(x) = 3x + 5.
+f(-10) = 3(-10) + 5 = -25
+### Step 6: Determine the final value of f(-10).
+Since the first equation gives us a valid value, we can conclude that f(-10) = -25.
+f(-10) = -25 and f(-10)"
+61,8,"The given equation is now $\frac{3\sqrt{27}}{1} = \frac{9}{2\sqrt{3}}$, and since $\sqrt{27} = \sqrt{9\cdot 3} = 3\sqrt{3}$, the equation is now $\frac{3\cdot 3\sqrt{3}}{1} = \frac{9}{2\sqrt{3}}$, and since $\frac{9}{2\sqrt{3}} = \frac{9\cdot \sqrt{3}}{2\sqrt{3}\cdot\sqrt{3}} = \frac{9\cdot\sqrt{3}}{2\cdot 3} = \frac{3}{2}$, the equation is now $\frac{9\sqrt{3}}{1} = \frac{3}{2}$, and since $\frac{9\sqrt{3}}{1} = 9\sqrt{3}$, the equation is now $9\sqrt{3} = \frac{3}{2}$, and since $\frac{3}{2} = 1.5$, the equation is now $9\sqrt{3} = 1.5$, and since $\sqrt{3} \approx 1.732$, the equation is now $9\cdot 1.732 = 1.5$, and since $9\cdot 1.732 = 15.588$, the equation is now $15.588 = 1.5$, and since $15.588 \approx 15.59$, the equation is now $15.59 = 1.5$, and since $15.59 = 10.38$, the equation is now $10.38 = 1.5$, and since $10.38 = 6.9$, the equation is now"
+62,187.5,"8 x 1.5 = 12, so a 12 fluid ounce bottle is 1.5 times larger than an 8 fluid ounce bottle. To find the calories in a 12 fluid ounce bottle, multiply the calories in an 8 fluid ounce bottle by 1.5. $125 \times 1.5 = \boxed{187.5}$.
+###
+Problem Goal:
+The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same common string in a jewelry store. Each string has a length of 30 inches. The following three strings are related by the same"
+63,\$40,"Five clusters can be made, which are (1, 2, 3, 4, 5), (1, 2, 4, 3, 5), (2, 3, 4, 1, 5), (1, 2, 3, 5, 4), and (2, 3, 5, 1, 4).
+Note: In this problem, we will assume that the order of the clusters does not matter. In other words, the cluster (1, 2, 3, 4, 5) is the same as (5, 1, 2, 3, 4).
+###
+Problem Goal:
+Given a set of integers $a$ and $b$ where $a$ is greater than $b$, the"
+64,5,"The value of $x$ is $\boxed{6}$.
+
+Note: The above steps are just one possible solution to the problem. There may be other possible next steps that could also lead to a solution."
+65,8,"Using the Pythagorean theorem, we can find the length of the hypotenuse by taking the square root of (5^2 + 12^2), which is 13. 
+###
+Problem Goal:
+A population of bacteria grows at a rate proportional to the number of bacteria present. At time t = 0, there are 100 bacteria, and at time t = 1, there are 120 bacteria. What is the rate constant for the growth of the bacteria?
+
+The bacteria grow at a rate proportional to the number of bacteria present. Let's assume the growth rate is r. At time t = 0, the rate of change is r * 100, and at time t = 1, the rate of change is r * 120. Since the bacteria grew from 100 to 120 in 1 time unit, the rate of change at time t = 0 is equal to the rate of change at time"
+66,3125,"To evaluate the expression $a^3\cdot a^2$, we can combine the exponents by adding them, which gives us $a^{3+2}=a^5$. Then, we can substitute the value of $a=5$ into the expression, resulting in $5^5=3125$."
+67,"[0,\infty)","The range of $f(x) = \sqrt{x^2}$ is $[0, \infty)$.
+###Note: The range of a function is the set of all possible output values it can produce for the input values in its domain. In this case, the function $f(x) = \sqrt{x^2}$ will always produce non-negative values, since the square root of a squared number is always non-negative. This is because the square root of a negative number is undefined in the real number system. Therefore, the range of the function is $[0, \infty)$."
+68,.5,"(x1+x2)=(-2+11/2)=(-4+11/2)=(-8+11)/2=(-1+11)/4=(-10+11)/4=(-1+1)/4=0.25
+###
+
+Note: This is a step-by-step guide on how to solve problems. The goal is to provide a clear and concise step-by-step process to solve a problem. The current state should provide a brief summary of the current understanding of the problem. The possible next step should provide a clear and concise next step to solve the problem. The goal is to make the problem easy to follow and understand. Please modify the examples to fit your specific problem."
+69,5,"The difference between 36 and 26 is $\boxed(10)$.
+
+### Problem Goal:
+Find the radius of the circle with equation $x^2 + 8x + y^"
+70,105,"The sum of the elements of each subset of the set {1,2,3,4,5,6} is $\boxed{231}$.
+Note: This is a very basic example. In a real-world scenario, the current state might be more complex, and the next step would require more detailed analysis and calculations."
+71,2,"Solve for n by rearranging the equation and plugging in the value of x. One way to do this is to take the log base 9 of both sides of the equation to get 18n = 24 log base 9 27. This can be further simplified to 18n = 24 log base 9 3^3. Using logarithm properties, this can be rewritten as 18n = 8 log base 9 3. Since log base 9 3 = 1/2, the equation can be simplified to 18n = 4. Solving for n, we get n = 4/9. However, the original problem statement is an equation of the form 9^x = 27^y. A more straightforward approach is to take the log base 9 of both sides and get x = 3y. Given that 9^x = 27^y, we can substitute 3 for x and y. Therefore, the value of n is 3. 
+Note: The original solution to the problem is indeed n = 3, but the step-by-step derivation is not provided. This is because the problem has been solved and there is no need to take further steps. However, the format requires a possible next step, and the current state is not sufficient to provide the solution. Therefore, the possible next step is to take the log base 9 of both sides and get x = 3y, and then solve for n."
+72,"(-\infty,-8)\cup (8,\infty)","Consider the discriminant, $\Delta=m^2-4(2)(8)=m^2-64$, and recall that a quadratic equation has two distinct real roots if and only if its discriminant is positive. Therefore, we can write an inequality in terms of $m$:
+$\Delta>0 \Rightarrow m^2-64>0 \Rightarrow (m-8)(m+8)>0 \Rightarrow (m<-8) \cup (m>8)$.
+###
+Problem Goal:
+Find the values of $x$ for which the function $f(x)=\sin\left(\frac{\pi}{4}+x\right)$ is positive.
+
+Since $\sin\left(\frac{\pi}{4}+x\right)=\sin\left(\frac{\pi}{4}\right)\cos(x)+\cos\left(\frac{\pi}{4}\right)\sin(x)$, we can use the angle addition formula for sine to simplify the expression for $f(x)$. Furthermore, since $\sin\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$ and $\cos\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$, we have that \begin{align*}
+f(x)&=\frac{1}{\sqrt{2}}\cos(x)+\frac{1}{\sqrt{2}}\sin(x)\\
+&=\frac{1}{\sqrt{2}}\left(\cos(x)+\sin(x)\right).
+\end"
+73,0,"Verify that the distribution is indeed optimal. One way to do this is to plot the profit for each type of product against the budget for each type of product. The optimal solution is the point on the graph where the profit is maximized.
+Note: This problem requires a graphical solution, which is not easily represented in text format. The goal of this problem is to encourage the solver to think creatively and consider different approaches to solve the problem. The solution is not unique and may vary depending on the specific values of X, Y, and Z."
+74,2,"The difference between 4817 and 4800 is $\boxed{17}$
+###
+
+
+Note: I've tried to follow the format as closely as possible, but please let me know if there's anything I can improve. Let me know if you have any further questions or if you'd like me to generate more examples!"
+75,\frac{1,"Simplify the fraction $\frac{1}{12}$.
+
+The fraction $\frac{1}{12}$ is already simplified. Reiterate the current state:
+The fraction $\frac{1}{12}$.
+
+Note: This is not a full solution to the problem, but a possible next step to solve the problem from the current state. The problem may require multiple steps to solve."
+76,6+9i,The expression $6+9i$ is indeed simplified. The final answer is $\boxed{6+9i}$.
+77,2,"Calculate the sums for both sequences using the formulas and find their difference.
+###
+## Step 1: Calculate the sum of the first 10 terms of the arithmetic sequence.
+## Step 2: This gives $S_{10} = \frac{10}{2}(2+12)=55$.
+## Step 3: Calculate the sum of the first 10 terms of the geometric sequence.
+## Step 4: This gives $S_{10} = \frac{2(1-\left(\frac{3}{2}\right)^{10})}{1-\frac{3}{2}}$.
+## Step 5: Simplify the expression for the geometric sequence sum.
+## Step 6: This gives $S_{10} = \frac{2(1-\frac{59049}{1024})}{-\frac{1}{2}}$.
+## Step 7: Simplify the expression further.
+## Step 8: This gives $S_{10} = -4(1-\frac{59049}{1024})$.
+## Step 9: Simplify the expression further.
+## Step 10: This gives $S_{10} = 4(\frac{1024-59049}{1024})$.
+## Step 11: Simplify the expression further.
+## Step 12: This gives $S_{10} = 4(\frac{-58025}{1024})$.
+## Step"
+78,20,"We can use the quadratic formula to solve for $x$: $$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$In our problem, $a=1$, $b=2$, and $c=-7$. Plugging these values into the formula, we get: $$x=\frac{-2\pm\sqrt{2^2-4\cdot1\cdot-7}}{2\cdot1}=\frac{-2\pm\sqrt{4+28}}{2}=\frac{-2\pm\sqrt{32}}{2}=\frac{-2\pm4\sqrt{2}}{2}=\frac{-2\pm4\sqrt{2}}{2}=\frac{-1\pm2\sqrt{2}}{1}$$Therefore, the solutions for $x$ are $x=1+2\sqrt{2}$ and $x=-1-2\sqrt{2}$.
+###
+Problem Goal:
+A set of four coins, each with a different denomination: 1c, 2c, 5c, and 10c, are flipped. What is the probability that the total"
+79,7(x+3) (x-3),"The expression $(x+3)(x+4)$ is equal to $x^2+7x+12$, which is not equal to $x^2-5x-12$. However, the expression $(x+4)(x-3)$ is equal to $x^2-x-12$. The expression $(x+3)(x-4)$ is equal to $x^2-x-12$. 
+Possible next steps:
+The expression $x^2+5x+12$ can be rewritten as $(x+3)(x+4)$, which equals $x^2+7x+12$. The expression $x^2+7x+12$ can be rewritten as $(x+3)(x+4)$. The expression $(x+3)(x+4)$ is equal to $x^2+7x+12$, which is not equal to $x^2-5x-"
+80,y^4-2y^3+7y^2+y-5,"Since the quadratic equation was solved, we will now plug the solutions into the original functions to check if they are valid.
+For the first solution, $x=\frac{-3 + \sqrt{17}}{4}$, we plug it into the function $f(y) = y^4 -3y^3 +y - 3$ and get $f(\frac{-3 + \sqrt{17}}{4}) = (\frac{-3 + \sqrt{17}}{4})^4 -3(\frac{-3 + \sqrt{17}}{4})^3 +(\frac{-3 + \sqrt{17}}{4}) - 3$.
+For the second solution, $x=\frac{-3 - \sqrt{17}}{4}$, we plug it into the function $f(y) = y^4 -3y^3 +y - 3$ and get $f(\frac{-3 - \sqrt{17}}{4}) = (\frac{-3 - \sqrt{17}}{4})^4 -3(\frac{-3 - \sqrt{17}}{4})^3 +(\frac{-3 - \sqrt{17}}{4}) - 3$.
+
+Note: Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the value of $f(y) + g(y)$, we should add the two solutions together.
+Since the problem asks for the"
+81,0,"The average of $x$ and $y$ can be rewritten as $\frac{x+y}{2}=\frac{x}{2}+\frac{y}{2}$.
+###
+Problem Goal:
+Suppose that $f$ is an invertible function from $(-\infty,\infty)$ to $(-\infty,\infty)$.  If $f(f(x))=x$, find $f(x)-f^{-1}(x)$.
+
+
+We can substitute $f(x)$ into the equation $f(f(x)) = x$ to get $f(x) = f^{-1}(x)$.
+
+## Step 3: Let's consider the given expression $x_{n} = f^{-1}(f^{-1}(...f^{-1}(x)...f^{-n}(...x)...))$, where $x$ is repeated $n$ times.
+Given the expression $x_{n} = f^{-1}(f^{-1}(...f^{-1}(x)...f^{-n}(...x)...))$, where $x$ is repeated $n$ times, we can see that this expression is a composition of $f^{-1}$ with itself $n$ times.
+
+## Step 4: We can simplify the expression $x_{n}$ by realizing that $f^{-1}(f^{-1}(x)) = x$.
+Since $f(f(x)) = x$, we can simplify the expression $x_{n}$ by realizing that $f^{-1}(f^{-1}(x)) = x$. This means that $x_{n}$ is equivalent to $x$.
+
+## Step 5: Now, we can substitute $x_{n} = x$ into the equation $y = f(x_{n})$.
+Substituting $x_{n} = x$ into the equation $y = f(x_{n})$, we get $y = f(x)$.
+
+## Step 6: Given that $f(f(x)) = x$, we can conclude that $f(x) = x + 1$.
+Since $f(f(x)) = x$, we can conclude that $f(x) = x + 1$.
+
+## Step 7: Now, we can find the value of $f(x_{n}) - f^{-1}(x_{n})$.
+Using the fact that $x_{n} = x$ and $f(x) = x + 1$, we can find the value of $f(x_{n}) - f^{-1}(x_{n})$. Since $f(x) = x + 1$, we have $f(x_{n}) = x_{n} + 1$. Also, $f^{-1}(x) = x - 1$. Therefore, $f(x_{n}) - f^{-1}(x_{n}) = (x_{n} + 1) - (x_{n} - 1) = 2$.
+
+The final answer is: $\boxed{2}$"
+82,\frac{7,(No calculation is needed. The expression is already simplified.)
+83,5,"We can use the general term of the sequence to find the 100th term, 23, by plugging in values for a_1, d, and n in the formula $a_n = a_1 + (n-1)d$.
+###
+Problem Goal:
+The"
+84,"\left(-\infty,-\frac 12\right)\cup \left(-\frac 12,\infty\right)","Since the current state is $(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)$, which already represents the domain of the function, there is no need to do further calculations. The domain is already found."
+85,\frac{2,The final answer is $\boxed{\frac{1}{2}}$.
+86,7,"Although the snail takes 20 days to climb 20 feet, the question asks for 18 feet, so it will take the snail 20 days to climb 18 feet and will be at 18 feet.
+###
+Please provide the problems and states, and I will solve them following the given format."
+87,20,"The final result is 20, which solves the problem. Reiterate the current state value.
+### Problem Goal:
+Simplify $(3-i)(6+2i)$.
+
+$(-i \cdot 6) = -6i$.
+$(-i \cdot 2i) = 2$.
+## Step 12:  Now we can combine the results.
+## Step 13:  The final result is $18 + 6i - 6i + 2 = 20$.
+
+The final answer is: $\boxed{20}$"
+88,9,"We can now find the derivatives of the numerator and denominator separately.
+Note: This is not a fully solved problem, but it is a good start. A next step could be to use the fact that the numerator and denominator are both polynomials of degree $2$, so that their derivatives can be found using the power rule.
+### Problem Goal:
+What is the value of $x$"
+89,4,"The value of the expression is $\sqrt[3]{2^{\frac{16}{3}}}$. 
+### 
+
+### 
+### 
+Let me provide the rest of the examples and the main problem.
+
+## Step 1: Problem Goal
+What is the value of $\left(16\sqrt[3]{2^7}\right)^{\frac 13}$?
+
+## Step 2: Current State
+We can start by evaluating the expression inside the parentheses: $\sqrt[3]{2^7}$.
+
+## Step 3: Possible next step
+The value of $\sqrt[3]{2^7}$ is $2^{\frac{7}{3}}$. We can now substitute this value into the original expression: $\left(16\cdot 2^{\frac{7}{3}}\right)^{\frac 13}$.
+
+## Step 4: Current State
+We can now simplify the expression: $\left(2^4\cdot 2^{\frac{7}{3}}\right)^{\frac 13} = \left(2^{\frac{22}{3}}\right)^{\frac 13} = 2^{\frac{22}{9}}$.
+
+## Step 5: Possible next step
+The value of the expression is $2^{\frac{22}{9}}$.
+
+The final answer is: $\boxed{2^{\frac{22}{9}}}$"
+90,6,The final answer is 6.
+91,"\left(1,\frac{9","Since 2^4 = 16, x = 4.
+Note that the next steps are not always correct and should"
+92,-2,"The value of h is indeed -2, so the current state is correct. The current state value is: $\boxed{(-2)}$.
+###"
+93,8,"If we plug in a=1, b=2, and c=-7 into the quadratic formula, we get x=\frac{-2\pm\sqrt{2^2-4(1)(-7)}}{2(1)}, which simplifies to x=\frac{-2\pm\sqrt{4+28}}{2}, which further simplifies to x=\frac{-2\pm\sqrt{32}}{2}, which is x=\frac{-2\pm4\sqrt{2}}{2}, which is x=\frac{-2+4\sqrt{2}}{2} or x=\frac{-2-4\sqrt{2}}{2}, which is x=-1+2\sqrt{2} or x=-1-2\sqrt{2}.
+###
+Note that the current state is not a simple numerical value, but rather an algebraic expression. However, in this case, the current state is still a valid solution to the problem, so we can reiterate it."
+94,20,"The final answer is 20, which is the greater of the two original numbers."
+95,24,"Because $j$ and $k$ are inversely proportional, we can set up a proportion $\frac{j}{k} = c$ for some constant $c$. Using the given values $j=16$ and $k=21$, we can find the constant $c$: $$\frac{j}{k} = c \Rightarrow \frac{16}{21} = c \Rightarrow c = \frac{8}{21}$$. Now, using this constant, we can find the new value of $j$: $$\frac{j}{14} = \frac{8}{21} \Rightarrow j = \frac{8}{21}\cdot 14 = \frac{8}{21}\cdot \frac{14}{1} = \frac{112}{21} = \frac{32}{7} = \boxed{\frac{32}{7}}$
+
+Note: The current state provided is not a solution, but rather an intermediate step."
+96,5,The sum of the possible values of $x$ is $\boxed{10}$.
+97,-7,Simplify the expression: $\frac{7}{2}$.
+98,0.43,"Rounded to the nearest hundredth, the ball will hit the ground after 0.37 seconds."
+99,108,The second horse needs to make 4 revolutions to travel the same distance as the first horse.
+100,2,"The plant will be 25 cm tall after 5 days.
+###
+Please let me know if you would like me to add more examples or if you have any further requests. 
+
+## Step 1: Understand the problem
+The problem presents a series of mathematical problems and asks for a possible next step to solve each problem from the current state.
+
+## Step 2: Analyze the problems
+The problems are a mix of different mathematical concepts, including percentages, rational expressions, and growth rates.
+
+## Step 3: Identify the next step for each problem
+For the first problem, the next step is to calculate the difference between 120% of 30 and 130% of 20. For the second problem, the next step is to check the results by plugging in x = -3 and x = 2 into the original function. For the third problem, the next step is to calculate the final height of the"
+101,10,"The difference between 36 and 26 is $\boxed{10}$.
+###
+Note: The original problem was removed as it was too long and did not fit the requested format. The example was kept to demonstrate the correct format. 
+
+The final answer is: $\boxed{10}$"
+102,i,The solution to the inequality is $x<\boxed{-3}$
+103,4,The final answer is: $\boxed{2}$
+104,402,"None, the current state is a solution.
+###
+
+
+## Step 1: Identify the problem goal and the current state.
+The problem goal is to find the positive difference between 2000 and 900, and the current state is that the difference is 1100.
+
+## Step 2: Check if the current state is a solution to the problem goal.
+The current state states that the difference between 2000 and 900 is 1100, which is indeed the positive difference.
+
+## Step 3: Since the current state is a solution, there is no need for a next step to solve the problem.
+
+The final answer is: $\boxed{1100}$"
+105,"x \in [-2,7]","Since the interval is $[-2, 7]$, we can write the solution in interval notation as $[-2, 7]$.
+
+The final answer is: $\boxed{[-2, 7]}$"
+106,-\frac{1,"The smallest value of $x$ is $-\frac{1}{8}$.
+### Problem Goal:
+What is the value of x that minimizes the function f(x) = x^2 + 2x - 6?
+
+
+To find the value of x that minimizes the function f(x) = x^2 + 2x - 6, we can complete the square or use calculus. Here, we'll use calculus.
+
+## Step 1: Find the derivative of f(x)
+To find the minimum value of the function, we need to find the critical points by taking the derivative of f(x) with respect to x. The derivative of f(x) = x^2 + 2x - 6 is f'(x) = 2x + 2.
+
+## Step 2: Set the derivative equal to zero
+To find the critical points, we set the derivative equal to zero: 2x + 2 = 0.
+
+## Step 3: Solve for x
+Solving the equation 2x + 2 = 0 gives x = -1.
+
+## Step 4: Check if x = -1 is a minimum
+To verify that x = -1 is a minimum, we can take the second derivative of f(x). The second derivative of f(x) is f''(x) = 2, which is positive for all x. This confirms that x = -1 is a minimum.
+
+## Step 5: Determine the minimum value of f(x)
+Substitute x = -1 into f(x) = x^2 + 2x - 6 to find the minimum value of the function: f(-1) = (-1)^2 + 2(-1) - 6 = 1 - 2 - 6 = -7.
+
+## Step 6: Since we found the value of x that minimizes the function, our final answer is x = -1.
+
+The final answer is: $\boxed{-1}$"
+107,\frac{x+2,"To find the inverse function $h^{-1}(x)$, we need to swap $x$ and $y$ and solve for $y$ in the equation $x = \frac{y + 2}{7}$.
+Next, multiply both sides by 7 to get $7x = y + 2$, then subtract 2 from both sides to get $7x - 2 = y$.
+Now we have the inverse function $h^{-1}(x) = 7x - 2$.
+###
+Problem Goal:
+A certain company's profit is $P$ dollars, and its cost is $C$ dollars. The company's profit is $P = 2000 - C$ dollars.
+
+The company's profit is $P = 2000 - C$ dollars.
+Next, set $P$ equal to $2000 - C$ to get the equation $P = 2000 - C$.
+Next, rearrange the equation to isolate $C$ by subtracting $P$ from both sides to get $C = 2000 - P$.
+Note: Rearranging the equation is not always the correct step, and in some cases, it may not be possible to isolate the variable.
+###
+Problem Goal:
+A certain type of bacteria doubles in number every hour. If a certain type of bacteria is initially present in a culture and the culture is left to incubate for 3 hours, how many bacteria will be present after 3 hours?
+
+The bacteria doubles every hour, and the culture is incubated for 3 hours.
+Next, calculate the number of bacteria present after 1 hour: $1\cdot 2 = 2$.
+Next, calculate the number of bacteria present after 2 hours: $2\cdot 2 = 4$.
+Next"
+108,10,"The distance between the origin and a point $(-8,6)$ is given by the formula $\sqrt{(-8)^2+6^2}=\sqrt{64+36}=\sqrt{100}=10.$
+The final answer is:"
+109,8,"The final answer is: $\boxed{8}$
+###
+Note that the final answer is already given, so there is no next step to propose."
+110,-25,"We need to check the second derivative to determine whether the critical point corresponds to a maximum or a minimum.
+$
+$
+The second derivative is $f''(x)=12x-10$. Plugging in $x=\frac{13}{12}$ gives $f''\left(\frac{13}{12}\right)=12\left(\frac{13}{12}\right)-10=13-10=3$, which is positive. This means that the critical point"
+111,8,"Since $A$ and $C$ have no common prime factors, $A$ and $C$ must be 3 and 2, respectively. Since $B$ has no perfect-square factors other than 1, $B$ must be 1. Therefore, $A + B + C = 3 + 1 + 2 = 6$.
+###
+Note: This is a typical example of a ""guess and check"" problem. If you get stuck, try guessing a possible solution, then verify it, and finally refine your guess. This is a good way to solve problems that are difficult to solve using traditional algebraic methods."
+112,187.5,The value of
+113,\$40,"Plug in $x=5$ into the equation: $y = 5(5)^2 + 3(5) - 2$, so $y = 5(25) + 15 - 2$, which simplifies to $y = 125 + 15 - 2$, which equals $y = 138$. The number of birds after 5 days is 138.
+Note that the next step is just a simple calculation that follows from the given information. In this case, the current state is actually the solution to the problem, so the next step is just to reiterate the current state value.
+Note that the next step is just a simple calculation that follows from the given information. In this case, the current state is actually the solution to the problem, so the next step is just to reiterate the current state value.
+Note that the next step is just a simple calculation that follows from the given information. In this case, the current state is actually the solution to the problem, so the next step is just to reiterate the current stamp offer price.
+Note"
+114,5,"The final answer is 5.
+### Note: The final answer is already provided in the example, so there is no need to calculate it. The task is to simply reiterate the final answer value."
+115,8,Try different values for P1 or V2 to see if any of the provided answer choices match the calculation.
+116,3125,"Set each factor equal to 0 and solve for $x$. This will give us the values of $x$ that satisfy the equation.
+###
+We will solve the equation by factoring.
+###
+Problem Goal:
+What is the value of $x"
+117,"[0,\infty)","None.
+###
+Problem Goal:
+What is the value of $x$ such that $x^2+2x-15=0$?
+
+The equation $x^2+2x-15=0$ can be factored into $(x+5)(x-3)=0$, and the solutions are $x=-5$ and $x=3$.
+Possible next"
+118,.5,"The final answer is $\boxed{All values of x}$.
+###
+Problem Goal:
+What is the value of $x$ in the equation $x^2 + 4x + 4 = (x + 2)^2$.
+
+The equation $x^2 + 4x + 4 = (x + 2)^2$ can be rewritten as $x^2 + 4x + 4 = x^2 + 4x + 4$.
+
+## Step 1: Expand the right side of the equation
+$(x + 2)^2 = x^2 + 4x + 4$
+
+## Step 2: Rewrite the equation
+$x^2 + 4x + 4 = x^2 + 4x + 4$
+
+##"
+119,5,"Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, we can plug in the values of $m$ and $(x_1, y_1)$ to get the equation of the line. Let's first calculate the slope $m$. $m = \frac{5 - 3}{4 - 2} = \frac{2}{2} = 1$.
+
+The final answer is: $\boxed{1}$"
+120,105,"No more steps needed, the current state already solves the problem."
+121,2,"Since $27=3^3$, we can write $27^{24}$ as $(3^3)^{24}=3^{3\cdot24}=3^{72}$.
+Now, since $9=3^2$, we can write $9^{18n}$ as $(3^2)^{18n}=3^{36n}$.
+So, $9^{18n}=27^{24}$ becomes $3^{36n}=3^{72}$.
+Since the bases are the same, we can set the exponents equal to each other: $36n=72$.
+Dividing both sides by 36, we get $n=\frac{72}{36}=\frac{2}{1}=2$.
+Therefore, the value of $n$ is $\boxed{2}$.
+###
+Problem Goal:
+A snail is at the bottom of a 20-foot well. Each day, it climbs up 3 feet, but at night, it slips back 2 feet. How many days will it take for the snail to reach the top of the well?
+
+After 2 days, the snail will have climbed 3 feet on the first day and 3 feet on the second day, for a total of 6 feet. However, on the second night, it will slip back 2 feet, so it will be at 4 feet.
+After 3 days, the snail will climb 3 feet on the third day and 3 feet on the fourth"
+122,"(-\infty,-8)\cup (8,\infty)","The values of $m$ that satisfy the inequality $|m|>8$ can be expressed in interval notation as $(-\infty, -8)\cup(8, \infty)$."
+123,0,"Substituting a = 3 and b = 4 into the equation $c^2 = a^2 + b^2$ yields $c^2 = 3^2 + 4^2$, which can be solved for c.
+###
+
+
+## Step 1: Understand the problem goal and the current state.
+The problem goal is to find the length of the hypotenuse of a right triangle with legs of length 3 and 4. The current"
+124,2,"Subtract 5 from both sides of the equation.
+###"
+125,\frac{1,Set each factor equal to zero and solve for x: $x + 3 = 0$ or $x + 2 = 0$. This gives us two possible solutions: $x = -3$ or $x = -2$.
+126,6+9i,"The final answer is $\boxed{12+9i}$.
+###
+Note: This is a very specific and simple problem, so the next steps are relatively easy to guess. As the problem gets harder, the next steps become harder to guess and the algorithm is more complex."
+127,2,"The expression $\frac{a+11b}{a-b}$ can be rewritten as $\frac{a-2b+9b}{a-2b}=\frac{a-2b}{a-2b}+9b$.
+###
+Note that the next step should be a mathematical expression or equation that is a natural next step from the current state, rather than a numerical answer. Also, please use LaTeX to format your answers. I'll be happy to help you with the rest of the problems. 
+
+Let me know when you're ready to submit the next problem, and I'll help you with the next step. 
+
+Please provide the next problem goal and the current state, and I'll help you propose a next step. 
+
+Also, please let me know if you want me to format the final answer as a box or not. 
+
+Looking forward to helping you solve the problems! 
+
+Best regards,
+[Your AI Assistant] 
+
+---
+
+Please provide the next problem goal and the current state, and I'll help you propose a next step. 
+
+You can format the problem goal and current state using LaTeX, if you want. 
+
+I'll be happy to help you with the rest of the problems. 
+
+Best regards,
+[Your AI Assistant] 
+
+---
+
+Please go ahead and provide the next problem goal and the current state. 
+
+I'm ready to help! 
+
+Best regards,
+[Your AI Assistant] 
+
+---
+
+Please provide the next problem goal and the current state, and I'll help you propose a next step. 
+
+You can format the problem goal and current state using LaTeX, if you want. 
+
+I'll be happy to help you with the"
+128,20,The product of the two complex numbers is $\boxed{(10-10i)}$.
+129,7(x+3) (x-3),"(Nothing to do, the problem is solved)."
+130,y^4-2y^3+7y^2+y-5,"The distance between 26 and 27 is $\boxed{1}$, and the distance between 28"
+131,0,"The equation $x^3+bx^2+2x+1=0$ has three real roots, $r_1$, $r_2$, and $r_3$.
+###
+Problem Goal:
+A certain plant grows from a seed and, in the first year, it grows to a height of 1 meter.  In each of the subsequent years, it grows by a factor of 1.1.  If the plant is 3.4 meters tall after 8 years, how many meters did it grow in the first year?
+
+We can write $1.1^8h=3.4$, where $h$ is the height of the plant in the first year.
+###
+Problem Goal:
+If $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$, find $\frac{x+y}{xy}$.
+
+We have
+\begin{align*}
+\frac{1}{x}+\frac{1}{y}&amp;=\frac{1}{z}\\
+\Rightarrow\qquad\qquad\frac{x+y}{xy}&amp;=\frac"
+132,\frac{7,"The answer can be simplified further by rationalizing the denominator. However, we can also rewrite $\sqrt{2}$ as $2^{1/2}$, which is equivalent to $\log_28^{1/2}$. Then, $\log_28\sqrt{2}=\log_28^{1/2}=\frac{1}{2}\log_28$. Since $\log_28=\frac{\log 28}{\log 8}$, we have $\log_28^{1/2}=\frac{1}{2}\cdot\frac{\log 28}{\log 8}=\frac{\log 28}{2\log 8}=\frac{\log 28}{2\log 2^3}=\frac{\log 28}{2\cdot 3\log 2}=\frac{\log 28}{6\log 2}=\frac{\log 2^3\cdot7}{6\log 2}=\frac{3\log 2+ \log 7}{6\log 2}=\frac{3}{6}+\frac{\log 7}{6\log 2}=\frac{1}{2}+\frac{\log 7}{6\log 2}$.
+### 
+
+Note: The final answer is already given in the prompt, so there is no need to calculate it. The task is to propose a possible next step. 
+
+Here is the revised response:
+
+ Problem Goal:
+Evaluate"
+133,5,"Since $\sqrt{8}=2\sqrt2$, we can substitute this into the original expression to obtain \[\log_{\sqrt8}(64\sqrt{8})=\log_{2\sqrt2}(64(2\sqrt2)).\]"
+134,"\left(-\infty,-\frac 12\right)\cup \left(-\frac 12,\infty\right)","The domain of the function $k(y) = \frac{1}{2y+1}$ can be expressed in interval notation as $(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)$.
+
+The final answer is: $\boxed{(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)}$"
+135,\frac{2,"The probability that for the $n$ I chose, there exist no real solutions to the equation $x(x+5) = -n$ is $\frac{1}{2}$.
+###
+Problem Goal:
+What is the value of $\left(\frac{3}{5}\right)^{\frac{7}{2}}$?
+
+To evaluate this expression, we will use the property $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
+
+## Step 1: Apply the property
+Using the property, we have $\left(\frac{3}{5}\right)^{\frac{7}{2}} = \frac{3^{\frac{7}{2}}}{5^{\frac{7}{2}}}$.
+
+## Step 2: Simplify the exponent
+We can simplify the exponent by breaking it down into two fractions: $\frac{7}{2} = \frac{4}{2} + \frac{3}{2}$. This gives $\frac{3^{\frac{7}{2}}}{5^{\frac{7}{2}}} = \frac{3^4 \cdot 3^{\frac{3}{2}}}{5^4 \cdot 5^{\frac{3}{2}}}$.
+
+## Step 3: Simplify the expression
+Now we can simplify the expression: $\frac{3^4 \cdot 3^{\frac{3}{2}}}{5^4 \cdot 5^{\frac{3}{2}}} = \frac{81 \cdot \sqrt{27}}{625 \cdot \sqrt{125}}$.
+
+## Step 4: Simplify the radicals
+We can simplify the radicals: $\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}$ and $\sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}$.
+
+## Step 5: Substitute the simplified radicals
+Substituting the simplified radicals, we have $\frac{81 \cdot 3\sqrt{3}}{625 \cdot 5\sqrt{5}}$.
+
+## Step 6: Simplify the fraction
+Now we can simplify the fraction: $\frac{81 \cdot 3\sqrt{3}}{625 \cdot 5\sqrt{5}} = \frac{243\sqrt{3}}{3125\sqrt{5}}$.
+
+## Step 7: Rationalize the denominator
+To rationalize the denominator, we can multiply both the"
+136,7,"Since $(x-1)^2$ is nonnegative for all real $x$, the equation has no solution.
+Note: This is a simplified response. The actual response should include a more detailed explanation of the steps taken to arrive at the solution. 
+
+Please provide the problem statement and the current state, and I will help you with the next step. 
+
+## Step 1: Understand the problem statement
+The problem statement is: ""Given that $f(x)$ is a function such that $f(1)=2$, $f(4)=3$, $f(7)=4$, and $f^{-1}(x)$ is the inverse of $f(x)$, what is $f^{-1}(f^{-1}(3))$?""
+The current state is ""$f^{-1}(4)$ is $\boxed{7}$.""
+
+## Step 2: Recall the definition of the inverse function
+The inverse function $f^{-1}(x)$ is a function that ""reverses"" the original function $f(x)$. This means that $f(f^{-1}(x))=x$ for all $x$ in the domain of $f$.
+
+## Step 3: Use the definition of the inverse function to find $f^{-1}(3)$
+Since $f("
+137,20,"The final answer is $\boxed{18 + 2i}$.
+
+Note that the problem is already solved, so we reiterate the current state."
+138,9,"We can verify that the answer is correct by using Vieta's formulae. For a quadratic equation $ax^2+bx+c=0$, the product of the solutions is $c/a$ and the sum of the solutions is $-b/a$. Therefore, the product of the squares of the solutions of $2x^2+13x+6=0$ is $(c/a)^2=(6/2)^2=9$."
+139,4,"This equation is a perfect square trinomial, so we can take the square root of both sides: $(x - 2)^2 = 0$ implies that $x - 2 = 0$, so $x = \boxed{2}.$
+###
+Problem Goal:
+What is the value of $x$ in the equation $x^3 - 2x^2 - 5x + 12 = 0$.
+
+We can factor this equation as $(x - 4"
+140,6,"The solutions to the quadratic equation are $x=-4$ and $x=1$. We will choose the positive solution, $x=1$.
+
+### Problem Goal:
+A snail is at the bottom of a 20-foot well. Each day, it climbs up 3 feet, but at night, it slips back 2 feet. How many days will it take for the snail to reach the top of the well?
+
+The snail climbs 3 feet during the day and slips back 2 feet at night, so it effectively moves 1 foot up the well each day.
+
+## Step 1: Calculate the number of days the snail needs to climb"
+141,"\left(1,\frac{9","If we choose $x=5$, we get $y=5\sqrt{3}$, but $y$ must be a positive integer. If we choose $x=6$, we get $y=4$, and we can check that $6^2+4^2=36+16=52\neq 100$. If we choose $x=8$, we get $y=6$, and we can check that $8^2+6^2=64+36=100$.
+Possible next step"
+142,-2,"The final answer is 2.
+###
+Problem Goal:
+The equation"
+143,8,"Divide both sides of the equation by 2 to solve for $x$.
+$2x = 11$
+$x = \frac{11}{2}$
+The final answer is $\boxed{5.5}$."
+144,20,"The lesser number is $3x$, which is $3(4)=12$. The ratio of the lesser number to the greater number after the operations is $\frac{12-4}{20+8} = \frac{8}{28}$, which simplifies to $\frac{2}{7}$. This matches the given condition, so the correct value for the greater number is indeed $20$. 
+
+The final answer is: $\boxed{40}$
+Note: The final answer is not 20, but one of the options given in the question. However, the correct answer is indeed 20.
+###
+The correct answer is indeed 20, but the problem statement asks for an answer from the options given. So, the correct next step is to re-evaluate the options given and find the one that matches the condition. In this case, the correct answer is 20, but it is not among the options given. Therefore, we need to re-evaluate the options given."
+145,24,"No change in the current state.
+### Note: The final answer is already given in the current state, so there is no need for a next step."
+146,5,The final answer is 5.
+147,-7,The final answer is 4.
+148,0.43,The ball hits the ground in $\boxed{0.37}$ seconds.
+149,108,"None, since the current state already solved the problem.
+### 
+Note: The final answer should be a number only. 
+Please keep the response in the format specified. 
+If you need to add anything else, let me know. 
+
+Since the current state already solved the problem, the possible next step is to simply reiterate the current state value.
+
+The final answer is: $\boxed{108}$"
diff --git a/run_bench.py b/run_bench.py
index cb3db6f..df6c2be 100644
--- a/run_bench.py
+++ b/run_bench.py
@@ -4,9 +4,11 @@ import argparse
 import time
 import vllm 
 from vllm import LLM, SamplingParams
-from transformers import AutoTokenizer, AutoModelForCausalLM
+from transformers import AutoTokenizer, AutoModelForCausalLM, TorchAoConfig
 import torch
 import torch.quantization
+# import evaluate
+from peft import AutoPeftModelForCausalLM
 
 from src.tot.data.benchmark.bench import *
 from src.tot.prompts.bench import value_prompt, propose_prompt
@@ -16,28 +18,63 @@ from torch.utils.data import DataLoader
 import random
 import multiprocessing
 
+import ast
+import re
+
+import pandas as pd
+
+import torchao
+from torchao.quantization.quant_api import (
+    quantize_,
+    int8_dynamic_activation_int8_weight,
+    int4_weight_only,
+    int8_weight_only
+)
+
+all_gt = []
+all_pred = []
+
+# total_runtime = []
+# average_sample_runtime = []
+# setup_time = []
+average_solving_time_per_sample = []
+average_proposal_time_per_sample = []
+average_eval_time_per_sample = []
+
+temp_tuning = {}
+
 def load_llama(quant=None):
     '''Load in one of the llama models'''
     tokenizer = AutoTokenizer.from_pretrained("meta-llama/Llama-3.2-3B-Instruct")
 
     if args.quantize and args.quantize=='ptq_int4':
-        model = AutoModelForCausalLM.from_pretrained("src/tot/quant/hf_quant_int4", device_map="cuda", weights_only=False)
+        quantization_config = TorchAoConfig("int4_weight_only", group_size=128)
+        model = AutoModelForCausalLM.from_pretrained("meta-llama/Llama-3.2-3B-Instruct", device_map="auto", quantization_config=quantization_config)
+        torchao.quantization.utils.recommended_inductor_config_setter()
+        # model = AutoModelForCausalLM.from_pretrained("src/tot/quant/hf_quant_int4", device_map="cuda", weights_only=False)
         model = torch.compile(model, mode="max-autotune")
     elif args.quantize and args.quantize=='ptq_int8':
-        model = AutoModelForCausalLM.from_pretrained("src/tot/quant/ptq_int8", device_map="cuda")
+        model = AutoModelForCausalLM.from_pretrained("meta-llama/Llama-3.2-3B-Instruct")
+        model.to('cuda')
+        quantize_(model, int8_weight_only())
+        # model = AutoModelForCausalLM.from_pretrained("src/tot/quant/ptq_int8", device_map="cuda", weights_only=False)
         model = torch.compile(model, mode="max-autotune")
     elif args.quantize and args.quantize == 'qat':
         model = AutoModelForCausalLM.from_pretrained("src/tot/quant/qat_int8", device_map="cuda", weights_only=False)
         model = torch.compile(model, mode="max-autotune")
-    elif args.vllm:
-        sampling_params = SamplingParams(n=1, max_tokens=100)
-        model = LLM(model="meta-llama/Llama-3.2-3B-Instruct", trust_remote_code=True, gpu_memory_utilization=0.9, max_model_len=2048)  # Name or path of your model
+    elif args.lora:
+        model = AutoPeftModelForCausalLM.from_pretrained("src/tot/lora")
+    # elif args.vllm:
+        # sampling_params = SamplingParams(n=1, max_tokens=100)
+        # model = LLM(model="meta-llama/Llama-3.2-3B-Instruct", trust_remote_code=True, gpu_memory_utilization=0.9, max_model_len=2048)  # Name or path of your model
     else:
         model = AutoModelForCausalLM.from_pretrained("meta-llama/Llama-3.2-3B-Instruct")
 
     return model, tokenizer
+def a_star_penalty(num, depth, k=0.1):
+    return num * np.exp(-k*depth)
 
-def value_proposals(problem, current_state, proposals, tokenizer, model, device):
+def value_proposals(problem, current_state, proposals, tokenizer, model, device, a_star=False, depth=None):
     '''
     Takes in string values of problem, current state, and proposals. 
     Returns a numerical valuation of each combination of the three factors above.
@@ -47,36 +84,125 @@ def value_proposals(problem, current_state, proposals, tokenizer, model, device)
     for p in proposals:
         prompts.append(value_prompt.format(problem=problem, current_state=current_state, proposal=p))
     
-    values = tokenizer(prompts, return_tensors='pt')
+    values = tokenizer(prompts, return_tensors='pt', padding=True, truncation=True)
     value_inputs = values['input_ids'].to(device)
     value_masks = values['attention_mask'].to(device)
 
     outputs = model.generate(value_inputs, attention_mask=value_masks, max_new_tokens=5)
     readable_outputs = tokenizer.batch_decode(outputs, skip_special_tokens=True)
-    for o in readable_outputs:
-        if 'sure' in o and 'current state is the solution' in o:
-            valuations.append(100.0)
-        elif 'sure' in o and 'current state is the solution' not in o:
-            valuations.append(1.0)
-        elif 'likely' in o:
-            valuations.append(0.5)
-        else:
-            valuations.append(0.0)
+    if not a_star:
+        for o in readable_outputs:
+            o = o.split("Evaluation:")[-1]
+            if 'sure' in o and 'current state is the solution' in o:
+                valuations.append(100.0)
+            elif 'sure' in o and 'current state is the solution' not in o:
+                valuations.append(1.0)
+            elif 'likely' in o:
+                valuations.append(0.5)
+            else:
+                valuations.append(0.0)
+    else:
+        for o in readable_outputs:
+            o = o.split("Evaluation:")[-1]
+            if 'sure' in o and 'current state is the solution' in o:
+                valuations.append(100.0)
+            elif 'sure' in o and 'current state is the solution' not in o:
+                valuations.append(a_star_penalty(1.0, depth=depth))
+            elif 'likely' in o:
+                valuations.append(a_star_penalty(0.5, depth=depth))
+            else:
+                valuations.append(0.0)
+
 
     return valuations
 
-def final_eval(gt, final_prop):
+def parse_problem(problem, math=False):
+    '''
+    parsing helper function
+    '''
+    if not math:
+        pattern = r"Choices:\s*(\[[^\]]+\])"
+
+        # Search for the pattern in the question string
+        match = re.search(pattern, problem)
+
+        # If there's a match, process the choices into a list
+        if match:
+            # Extract the matched string (the list as a string)
+            choices_str = match.group(1)
+            
+            # Convert the string into a Python list using ast.literal_eval
+            choices_list = ast.literal_eval(choices_str)
+            
+            # print(choices_list)
+
+            return choices_list
+        else:
+            print("No choices found.")
+            return []
+    else:
+        # print("THIS IS PROB")
+        # print(problem)
+        matches = re.findall(r'\\boxed{([^}]*)}', problem)
+        # print("THESE ARE MATCHES")
+        # print(matches)
+        if matches and len(matches) > 0:
+            return matches[-1]
+        else:
+            print("No choices found.")
+            return []
+
+
+def final_eval(gt, final_prop, problem):
     '''
     compare the ground truth and final proposed solution by the model
     '''
-    print("THIS IS THE FINAL PROP")
-    print(final_prop)
-    print("THIS IS THE GT")
-    
-    if gt in final_prop:
-        return 1.0
+    if "Current State" in final_prop:
+        final_prop = final_prop.split("Current_State")[-1]
+    final_prop = final_prop.replace("Possible next step:", "").replace("Current State:", "").strip()
+
+    if "Choices" in problem: #multiple choice scenario
+        try:
+            parsed = parse_problem(problem)
+            gt = parsed[int(gt)]
+            
+            # print(f"""
+            # ###
+            
+            # Here is the gt: {gt}
+
+            # Here is the final prop: {final_prop}
+
+            # ###
+            # """)
+
+            all_pred.append(final_prop)
+            all_gt.append(gt)
+            if gt in final_prop:
+                return 1.0
+            else: #other problem scenarios
+                return 0.0
+        except:
+            return 0.0
     else:
-        return 0.0
+        # print(gt)
+        gt = parse_problem(gt, math=True)
+        all_pred.append(final_prop)
+        all_gt.append(gt)
+        # print(f"""
+        # ###
+        
+        # Here is the gt: {gt}
+
+        # Here is the final prop: {final_prop}
+
+        # ###
+        # """)
+        if isinstance(gt, str) and gt in final_prop:
+            return 1.0
+        else:
+            return 0.0
+
 
 def get_test_data(tokenizer, batch_size):
     '''
@@ -101,11 +227,12 @@ def get_test_data(tokenizer, batch_size):
     mmlu_questions_proc_test = format_for_mm(mmlu_raw['test']['question'], mmlu_raw['test']['choices'])
 
     #master list - test
-    sublist_input_test = gpqa_questions_proc[158:] + math_raw['test']['problem'] + mmlu_questions_proc_test
-    sublist_answer_test = gpqa_raw['train']['Correct Answer'][158:] + math_raw['test']['solution'] + mmlu_raw['test']['answer']
-    agg_test_set = benchmark_dataset(sublist_input_test, sublist_answer_test, tokenizer)
+    # sublist_input_test = gpqa_questions_proc[158:] + math_raw['test']['problem'] + mmlu_questions_proc_test
+    # sublist_answer_test = gpqa_raw['train']['Correct Answer'][158:] + math_raw['test']['solution'] + mmlu_raw['test']['answer']
+    # agg_test_set = benchmark_dataset(sublist_input_test, sublist_answer_test, tokenizer)
+    agg_test_set = benchmark_dataset(math_raw['test']['problem'], math_raw['test']['solution'], tokenizer)
 
-    return DataLoader(agg_test_set, batch_size=batch_size, shuffle=True, collate_fn=collate_fn_qat)
+    return DataLoader(agg_test_set, batch_size=batch_size, collate_fn=collate_fn_qat)
 
 def solve(input_ids, label, mask, model, tokenizer, device, args):
     '''
@@ -118,26 +245,33 @@ def solve(input_ids, label, mask, model, tokenizer, device, args):
     for i in range(args.depth): #args.depth number of attempts to reach the solution
         
         #propose next step/solutions per node/prompt
-
+        rpropose = time.perf_counter()
         out = model.generate(
             input_ids,
             attention_mask=mask,
             temperature=args.temperature, 
-            max_new_tokens=args.max_new_tokens, 
+            max_new_tokens=input_ids.shape[1]+args.max_new_tokens, 
             num_return_sequences=args.breadth,
             )
+        rpropose = time.perf_counter()-rpropose
+        average_proposal_time_per_sample.append(rpropose)
 
-        
         #evaluate/rate the proposals
         current_state = tokenizer.batch_decode(input_ids, skip_special_tokens=True)[0]
 
         proposals = []
         for o in out:
-            string_answer = tokenizer.decode(o[-args.max_new_tokens:], skip_special_tokens=True)
-            assert isinstance(string_answer, str)
+            string_answer = tokenizer.decode(o, skip_special_tokens=True)
+            string_answer = string_answer.split("Possible next step:")[-1]
+            # print(string_answer)
+            # print("+++++"*50)
+            # assert isinstance(string_answer, str)
             proposals.extend([string_answer])
-
+        # exit()
+        reval = time.perf_counter()
         valuations = value_proposals(problem=problem, current_state=current_state, proposals=proposals, tokenizer=tokenizer, model=model, device=device)
+        reval = time.perf_counter() - reval
+        average_eval_time_per_sample.append(reval)
 
         #if the model believes it has reached the final solution before args.depth is up, break
         if 100.0 in valuations:
@@ -148,9 +282,6 @@ def solve(input_ids, label, mask, model, tokenizer, device, args):
         val_props.sort(key = lambda ele: ele[1], reverse=True)
         selected = val_props[:args.greedy_n][0][0]
 
-        # print("THIS IS SELCTED")
-        # print(selected)
-
         #format the chosen proposal for the next iteration
         next_prompt = propose_prompt.format(problem=problem, current_state=selected)
         inputs = tokenizer(next_prompt, return_tensors='pt')
@@ -160,7 +291,7 @@ def solve(input_ids, label, mask, model, tokenizer, device, args):
 
     #compare the proposed final answer vs the ground truth
     gt = tokenizer.batch_decode(label, skip_special_tokens=True)
-    judgement = final_eval(gt[0], selected)
+    judgement = final_eval(gt[0], selected, problem)
 
     return judgement
 
@@ -168,6 +299,11 @@ def run(args):
     '''
     main run function
     '''
+    torch.cuda.reset_peak_memory_stats()
+    torch.cuda.empty_cache()
+
+    rtotal = time.perf_counter()
+    rsetup = time.perf_counter()
     ### SETUP MODEL ###
     #bc of the way the original repo is structured, will need to load in llama models in run.py to avoid repeated loading in models.py
     if args.backend == 'llama':
@@ -179,6 +315,8 @@ def run(args):
         model = None
         tokenizer = None
 
+    tokenizer.pad_token = tokenizer.eos_token
+
     ### SETUP DATA ###
     test_data = get_test_data(tokenizer, args.concurrent)
 
@@ -188,6 +326,11 @@ def run(args):
     total = 0
     right = 0
 
+    custom_stop = args.num_test_samp if args.num_test_samp else 13706
+    count = 0
+
+    rsetup = time.perf_counter()-rsetup
+    
     for sample in test_data:
 
         #extract out the sample parts for the initial input
@@ -197,15 +340,61 @@ def run(args):
         
         #cannot get multiple gpus. will run this on a single gpu one sample at a time for simplicity
         for i in range(len(input_ids)):
+
+            rsolve = time.perf_counter()
             judgement = solve(input_ids[i].view(1,-1), label[i].view(1,-1), mask[i].view(1,-1), model, tokenizer, device, args)
+            rsolve = time.perf_counter()-rsolve
+            average_solving_time_per_sample.append(rsolve)
+
             total += 1.0
             right += judgement
-
+            count += 1
+            if count == custom_stop:
+                break
+        if count == custom_stop:
+            break
         #keep track of the running totals
         print("Accuracy so far: ", right/total)
 
     print("FINAL ACCURACY: ", right/total)
+    # temp_tuning[args.temperature] = right/total
 
+    #temp save
+    res = pd.DataFrame({
+        "gt": all_gt,
+        "pred": all_pred
+    })
+    if args.quantize:
+        res.to_csv(f"./results_{args.backend}_{args.quantize}_{args.temperature}_{args.num_test_samp}.csv")
+    elif args.lora:
+        res.to_csv(f"./results_{args.backend}_lora_{args.temperature}_{args.num_test_samp}.csv")
+    else:
+        res.to_csv("./res.csv")
+    
+    rtotal = time.perf_counter()-rtotal
+
+    peak = torch.cuda.max_memory_allocated()
+
+    time_df = pd.DataFrame({
+        "total_accuracy": right/total,
+        "total runtime": rtotal,
+        "total setup time": rsetup,
+        "average solving time": sum(average_solving_time_per_sample)/len(average_solving_time_per_sample),
+        "average proposal time": sum(average_proposal_time_per_sample)/len(average_proposal_time_per_sample),
+        "average eval time": sum(average_eval_time_per_sample)/len(average_eval_time_per_sample),
+        "peak memory usage": peak
+    }, index=[0])
+    if args.quantize:
+        time_df.to_csv(f"./times_{args.backend}_{args.quantize}_{args.temperature}_{args.num_test_samp}.csv")
+    elif args.lora:
+        time_df.to_csv(f"./times_{args.backend}_lora_{args.temperature}_{args.num_test_samp}.csv")
+    else:
+        time_df.to_csv("./times.csv")
+    
+    #courtesy prints
+    print("TOTAL RUNNING TIME: ", rtotal)
+    print("SETUP TIME: ", rsetup)
+    print(f"PEAK GPU MEM USAGE: {peak / 1e6:.2f} MB")
         
 
 def parse_args():
@@ -223,7 +412,9 @@ def parse_args():
     args.add_argument('--breadth', type=int, default=3)
     args.add_argument('--greedy_n', type=int, default=1)
     args.add_argument('--concurrent', type=int, default=4)
-    args.add_argument('--vllm', type=bool, default=False)
+    args.add_argument('--a_star', action='store_true')
+    args.add_argument('--lora', action='store_true')
+    args.add_argument('--num_test_samp', type=int)
 
     args = args.parse_args()
     return args
@@ -231,5 +422,33 @@ def parse_args():
 
 if __name__ == '__main__':
     args = parse_args()
+
+    #test base instruct llama
     print(args)
-    run(args)
\ No newline at end of file
+    run(args)
+
+    #test quant llama w/ qat int8
+    # args.quantize="qat"
+    # print(args)
+    # run(args)
+
+    #test llama w/ ptq int4
+    args.quantize="ptq_int4"
+    print(args)
+    run(args)
+
+    #test llama w/ ptq int8
+    args.quantize="ptq_int8"
+    print(args)
+    run(args)
+
+    #test llama w/ lora
+    # args.quantize=None
+    # args.lora = True
+    # print(args)
+    # run(args)
+
+    # print("THIS IS TEMP TUNING")
+    # print(temp_tuning.items())
+    # temp = pd.DataFrame(temp_tuning)
+    # temp.to_csv('./temp_tuning.csv')
\ No newline at end of file
diff --git a/src/tot/data/benchmark/bench.py b/src/tot/data/benchmark/bench.py
index e2e2309..8d45365 100644
--- a/src/tot/data/benchmark/bench.py
+++ b/src/tot/data/benchmark/bench.py
@@ -6,6 +6,7 @@ import random
 import multiprocessing
 import os
 from transformers import AutoTokenizer, AutoModelForCausalLM
+import re
 
 tokenizer = AutoTokenizer.from_pretrained("meta-llama/Llama-3.2-3B-Instruct")
 tokenizer.pad_token = tokenizer.eos_token
@@ -28,6 +29,11 @@ class benchmark_dataset(torch.utils.data.Dataset):
     new_label = []
 
     for q, a in zip(self.input, self.labels):
+
+      matches = re.findall(r'\\boxed{([^}]*)}', a)
+      if len(matches) <= 0:
+        continue
+
       tk_len_q = len(tokenizer(str(q), return_tensors='pt')['input_ids'][0])
       tk_len_a = len(tokenizer(str(a), return_tensors='pt')['input_ids'][0])
 
@@ -41,7 +47,7 @@ class benchmark_dataset(torch.utils.data.Dataset):
     Len of New_Input: {len(new_input)}
     Len of New_Label: {len(new_label)}
 
-    Sample Input, Label: {new_input[0], new_label[0]}
+    Sample Input, Label: {new_input[1], new_label[1]}
 
     """)
 
diff --git a/src/tot/lora/adapter_config.json b/src/tot/lora/adapter_config.json
new file mode 100644
index 0000000..49aef5c
--- /dev/null
+++ b/src/tot/lora/adapter_config.json
@@ -0,0 +1,29 @@
+{
+  "alpha_pattern": {},
+  "auto_mapping": null,
+  "base_model_name_or_path": "meta-llama/Llama-3.2-3B-Instruct",
+  "bias": "none",
+  "fan_in_fan_out": false,
+  "inference_mode": true,
+  "init_lora_weights": true,
+  "layer_replication": null,
+  "layers_pattern": null,
+  "layers_to_transform": null,
+  "loftq_config": {},
+  "lora_alpha": 32,
+  "lora_dropout": 0.1,
+  "megatron_config": null,
+  "megatron_core": "megatron.core",
+  "modules_to_save": null,
+  "peft_type": "LORA",
+  "r": 8,
+  "rank_pattern": {},
+  "revision": null,
+  "target_modules": [
+    "v_proj",
+    "q_proj"
+  ],
+  "task_type": "CAUSAL_LM",
+  "use_dora": false,
+  "use_rslora": false
+}
\ No newline at end of file
diff --git a/src/tot/lora/adapter_model.safetensors b/src/tot/lora/adapter_model.safetensors
new file mode 100644
index 0000000..b7dcffb
Binary files /dev/null and b/src/tot/lora/adapter_model.safetensors differ
diff --git a/src/tot/prompts/bench.py b/src/tot/prompts/bench.py
index 5bb5d6f..803436a 100644
--- a/src/tot/prompts/bench.py
+++ b/src/tot/prompts/bench.py
@@ -1,76 +1,31 @@
-propose_prompt = '''[INST]Given a problem goal and a current state, propose one possible next step to solve the problem from the current state. If the current state solves the problem, say "current state is the solution".
+propose_prompt = '''Given a latex problem goal and a current state, propose one possible next step to solve the problem from the current state. If the current state solves the problem, simply reiterate the current state value.
 ###
-Here are two examples to help:
-
-Example #1:
+Here is an example to help:
 Problem Goal:
-I have four numbers: 4, 4, 6, and 8. How can I use those numbers with basic arithmetic operations (+ - * /) to obtain 24?
+What is the positive difference between $120\\%$ of 30 and $130\\%$ of 20?
 Current State:
-4+8 = 12
-I have 4, 6, and 12 left.
+One hundred twenty percent of 30 is $120\\cdot30\\cdot\\(1/100)=36$, and $130\\%$ of 20 is $ 130\\cdot 20\\cdot\\(1/100)=26$.
 Possible next step:
-6-4 = 2
-
-Example #2:
-Problem Goal:
-Choose the choice that best answer the following question:
-    Question:
-    Davis decided to kill Adams. He set out for Adams's house. Before he got there he saw Brooks, who resembled Adams. Thinking that Brooks was Adams, Davis shot at Brooks. The shot missed Brooks but wounded Case, who was some distance away. Davis had not seen Case. In a prosecution under a statute that proscribes any attempt to commit murder, the district attorney should indicate that the intended victim(s) was/were
-    Choices:
-    ['Adams only.', 'Brooks only.', 'Case only.', 'Adams and Brooks']
-Current State:
-Choice 2
-Possible next step:
-current state is the solution.
-
+The difference between 36 and 26 is $\\boxed(10)$.
 ###
 Problem Goal:
 {problem}
 Current State:
 {current_state}
-Possible next step:[\INST]
+Possible next step:
 '''
 
-value_prompt = '''[INST]Given a problem goal, a current state, and a proposed next step, evaluate if the next step from the current state can help solve or answer the problem (yes/likely/no)
+value_prompt = '''Given a problem goal, a current state, and a proposed next step, evaluate if the next step from the current state can help eventually solve or answer the problem (yes/likely/no)
 ###
-Here are three examples to help:
-Example #1:
-Problem Goal: 
-I have four numbers: 4, 4, 6, and 8. How can I use those numbers with basic arithmetic operations (+ - * /) to obtain 24?
+Here is an example to help:
+Problem Goal:
+What is the positive difference between $120\\%$ of 30 and $130\\%$ of 20?
 Current State:
-4+8 = 12 (remaining numbers: 4, 6, 12)
-Proposal: 
-6-4 = 2 (remaining numbers: 2, 12)
+One hundred twenty percent of 30 is $120\\cdot30\\cdot\\frac(1/100)=36$.
+Possible next step:
+$130\\%$ of 20 is $130\\cdot 20\\cdot\\frac(1/100)=26$
 Evaluation:
 likely
-
-Example #2:
-Problem Goal:
-Choose the choice that best answer the following question:
-    Question:
-    Davis decided to kill Adams. He set out for Adams's house. Before he got there he saw Brooks, who resembled Adams. Thinking that Brooks was Adams, Davis shot at Brooks. The shot missed Brooks but wounded Case, who was some distance away. Davis had not seen Case. In a prosecution under a statute that proscribes any attempt to commit murder, the district attorney should indicate that the intended victim(s) was/were
-    Choices:
-    ['Adams only.', 'Brooks only.', 'Case only.', 'Adams and Brooks']
-Current State:
-Choice 4
-Proposal:
-current state is the solution.
-Evaluation:
-no
-
-Example #3:
-Choose the choice that best answer the following question:
-    Question:
-    Davis decided to kill Adams. He set out for Adams's house. Before he got there he saw Brooks, who resembled Adams. Thinking that Brooks was Adams, Davis shot at Brooks. The shot missed Brooks but wounded Case, who was some distance away. Davis had not seen Case. In a prosecution under a statute that proscribes any attempt to commit murder, the district attorney should indicate that the intended victim(s) was/were
-    Choices:
-    ['Adams only.', 'Brooks only.', 'Case only.', 'Adams and Brooks']
-Current State:
-Choice 2
-Proposal:
-current state is the solution.
-Evaluation:
-yes
-
 ###
 Problem Goal:
 {problem}
@@ -78,5 +33,88 @@ Current State:
 {current_state}
 Proposal:
 {proposal}
-Evaluation:[\INST]
-'''
\ No newline at end of file
+Evaluation:
+'''
+
+# propose_prompt = '''Given a problem goal and a current state, propose one possible next step to solve the problem from the current state. If the current state solves the problem, say "current state is the solution".
+# ###
+# Here are two examples to help:
+
+# Example #1:
+# Problem Goal:
+# I have four numbers: 4, 4, 6, and 8. How can I use those numbers with basic arithmetic operations (+ - * /) to obtain 24?
+# Current State:
+# 4+8 = 12
+# I have 4, 6, and 12 left.
+# Possible next step:
+# 6-4 = 2
+
+# Example #2:
+# Problem Goal:
+# Choose the choice that best answer the following question:
+#     Question:
+#     Davis decided to kill Adams. He set out for Adams's house. Before he got there he saw Brooks, who resembled Adams. Thinking that Brooks was Adams, Davis shot at Brooks. The shot missed Brooks but wounded Case, who was some distance away. Davis had not seen Case. In a prosecution under a statute that proscribes any attempt to commit murder, the district attorney should indicate that the intended victim(s) was/were
+#     Choices:
+#     ['Adams only.', 'Brooks only.', 'Case only.', 'Adams and Brooks']
+# Current State:
+# Brooks only.
+# Possible next step:
+# current state is the solution. Brooks only.
+
+# ###
+# Problem Goal:
+# {problem}
+# Current State:
+# {current_state}
+# Possible next step:
+# '''
+
+# value_prompt = '''Given a problem goal, a current state, and a proposed next step, evaluate if the next step from the current state can help solve or answer the problem (yes/likely/no)
+# ###
+# Here are three examples to help:
+# Example #1:
+# Problem Goal: 
+# I have four numbers: 4, 4, 6, and 8. How can I use those numbers with basic arithmetic operations (+ - * /) to obtain 24?
+# Current State:
+# 4+8 = 12 (remaining numbers: 4, 6, 12)
+# Proposal: 
+# 6-4 = 2 (remaining numbers: 2, 12)
+# Evaluation:
+# likely
+
+# Example #2:
+# Problem Goal:
+# Choose the choice that best answer the following question:
+#     Question:
+#     Davis decided to kill Adams. He set out for Adams's house. Before he got there he saw Brooks, who resembled Adams. Thinking that Brooks was Adams, Davis shot at Brooks. The shot missed Brooks but wounded Case, who was some distance away. Davis had not seen Case. In a prosecution under a statute that proscribes any attempt to commit murder, the district attorney should indicate that the intended victim(s) was/were
+#     Choices:
+#     ['Adams only.', 'Brooks only.', 'Case only.', 'Adams and Brooks']
+# Current State:
+# Adams and Brooks
+# Proposal:
+# current state is the solution. Adams and Brooks.
+# Evaluation:
+# no
+
+# Example #3:
+# Choose the choice that best answer the following question:
+#     Question:
+#     Davis decided to kill Adams. He set out for Adams's house. Before he got there he saw Brooks, who resembled Adams. Thinking that Brooks was Adams, Davis shot at Brooks. The shot missed Brooks but wounded Case, who was some distance away. Davis had not seen Case. In a prosecution under a statute that proscribes any attempt to commit murder, the district attorney should indicate that the intended victim(s) was/were
+#     Choices:
+#     ['Adams only.', 'Brooks only.', 'Case only.', 'Adams and Brooks']
+# Current State:
+# Brooks only.
+# Proposal:
+# current state is the solution. Brooks only.
+# Evaluation:
+# yes
+
+# ###
+# Problem Goal:
+# {problem}
+# Current State:
+# {current_state}
+# Proposal:
+# {proposal}
+# Evaluation:
+# '''
\ No newline at end of file
diff --git a/src/tot/quant/qat_int8_5/config.json b/src/tot/quant/qat_int8_5/config.json
new file mode 100644
index 0000000..135e945
--- /dev/null
+++ b/src/tot/quant/qat_int8_5/config.json
@@ -0,0 +1,40 @@
+{
+  "_name_or_path": "meta-llama/Llama-3.2-3B-Instruct",
+  "architectures": [
+    "LlamaForCausalLM"
+  ],
+  "attention_bias": false,
+  "attention_dropout": 0.0,
+  "bos_token_id": 128000,
+  "eos_token_id": [
+    128001,
+    128008,
+    128009
+  ],
+  "head_dim": 128,
+  "hidden_act": "silu",
+  "hidden_size": 3072,
+  "initializer_range": 0.02,
+  "intermediate_size": 8192,
+  "max_position_embeddings": 131072,
+  "mlp_bias": false,
+  "model_type": "llama",
+  "num_attention_heads": 24,
+  "num_hidden_layers": 28,
+  "num_key_value_heads": 8,
+  "pretraining_tp": 1,
+  "rms_norm_eps": 1e-05,
+  "rope_scaling": {
+    "factor": 32.0,
+    "high_freq_factor": 4.0,
+    "low_freq_factor": 1.0,
+    "original_max_position_embeddings": 8192,
+    "rope_type": "llama3"
+  },
+  "rope_theta": 500000.0,
+  "tie_word_embeddings": true,
+  "torch_dtype": "float32",
+  "transformers_version": "4.46.3",
+  "use_cache": true,
+  "vocab_size": 128256
+}
diff --git a/src/tot/quant/qat_int8_5/generation_config.json b/src/tot/quant/qat_int8_5/generation_config.json
new file mode 100644
index 0000000..aba45c8
--- /dev/null
+++ b/src/tot/quant/qat_int8_5/generation_config.json
@@ -0,0 +1,12 @@
+{
+  "bos_token_id": 128000,
+  "do_sample": true,
+  "eos_token_id": [
+    128001,
+    128008,
+    128009
+  ],
+  "temperature": 0.6,
+  "top_p": 0.9,
+  "transformers_version": "4.46.3"
+}
diff --git a/times.csv b/times.csv
new file mode 100644
index 0000000..276f29f
--- /dev/null
+++ b/times.csv
@@ -0,0 +1,2 @@
+,total_accuracy,total runtime,total setup time,average solving time,average proposal time,average eval time,peak memory usage
+0,0.38,3382.8438607280004,17.842933223000728,67.29928937723955,20.571769750779882,1.8586646483599665,14634687488
diff --git a/times_llama.csv b/times_llama.csv
new file mode 100644
index 0000000..2ed22f9
--- /dev/null
+++ b/times_llama.csv
@@ -0,0 +1,2 @@
+,total runtime,total setup time,average solving time,average proposal time,average eval time,peak memory usage
+0,1609.2750769329978,26.220256619999418,31.66035637013985,8.027884644520284,2.522419844319835,15983939584
diff --git a/times_llama_lora_0.7_50.csv b/times_llama_lora_0.7_50.csv
new file mode 100644
index 0000000..739b61e
--- /dev/null
+++ b/times_llama_lora_0.7_50.csv
@@ -0,0 +1,2 @@
+,total_accuracy,total runtime,total setup time,average solving time,average proposal time,average eval time,peak memory usage
+0,0.0,214.61139552399982,45.25130091099709,36.097559108329804,10.088780836261673,1.9391463807286102,14179190784
diff --git a/times_llama_ptq_int4_0.7_50.csv b/times_llama_ptq_int4_0.7_50.csv
new file mode 100644
index 0000000..f960169
--- /dev/null
+++ b/times_llama_ptq_int4_0.7_50.csv
@@ -0,0 +1,2 @@
+,total_accuracy,total runtime,total setup time,average solving time,average proposal time,average eval time,peak memory usage
+0,0.42,2503.1283878510003,16.094477847000235,58.519657035349844,17.752996491656535,1.7502601381599743,3191929856
diff --git a/times_llama_ptq_int8_0.7_50.csv b/times_llama_ptq_int8_0.7_50.csv
new file mode 100644
index 0000000..272a177
--- /dev/null
+++ b/times_llama_ptq_int8_0.7_50.csv
@@ -0,0 +1,2 @@
+,total_accuracy,total runtime,total setup time,average solving time,average proposal time,average eval time,peak memory usage
+0,0.36,4740.312165042,15.272982416998275,70.51314375465326,21.932249810273213,1.5675949395555493,12986608128
diff --git a/times_llama_qat_0.7_50.csv b/times_llama_qat_0.7_50.csv
new file mode 100644
index 0000000..0295012
--- /dev/null
+++ b/times_llama_qat_0.7_50.csv
@@ -0,0 +1,2 @@
+,total_accuracy,total runtime,total setup time,average solving time,average proposal time,average eval time,peak memory usage
+0,0.0,627.7139296419919,23.9375716640061,12.074782435418165,1.5538430808399184,2.4684065076932895,5936145408
diff --git a/times_ptqint4.csv b/times_ptqint4.csv
new file mode 100644
index 0000000..ed4eb7d
--- /dev/null
+++ b/times_ptqint4.csv
@@ -0,0 +1,2 @@
+,total runtime,total setup time,average solving time,average proposal time,average eval time,peak memory usage
+0,2320.7337444990007,25.463927106000483,45.904669674180184,12.689160688306535,2.6089166288600607,4242105856