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logs/game24/gpt-4o_0.7_propose5_value1_greedy1_start10_end15.json
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@ -8,7 +8,7 @@
""
],
"new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n",
"\n",
"1. \\(1 + 1 = 2\\) (left: 2, 2, 8)\n",
"2. \\(1 + 2 = 3\\) (left: 1, 3, 8)\n",
@ -17,14 +17,13 @@
"5. \\(1 \\times 2 = 2\\) (left: 1, 2, 8)\n",
"6. \\(1 \\times 8 = 8\\) (left: 1, 2, 8)\n",
"7. \\(2 + 8 = 10\\) (left: 1, 1, 10)\n",
"8. \\(8 - 1 = 7\\) (left: 1, 2, 7)\n",
"9. \\(8 - 2 = 6\\) (left: 1, 1, 6)\n",
"10. \\(2 - 1 = 1\\) (left: 1, 1, 8)\n",
"11. \\(8 / 1 = 8\\) (left: 1, 2, 8) \n",
"12. \\(8 / 2 = 4\\) (left: 1, 1, 4)\n",
"13. \\(2 / 1 = 2\\) (left: 1, 1, 8)\n",
"8. \\(2 - 1 = 1\\) (left: 1, 1, 8)\n",
"9. \\(8 - 1 = 7\\) (left: 1, 2, 7)\n",
"10. \\(8 - 2 = 6\\) (left: 1, 1, 6)\n",
"11. \\(8 \\div 1 = 8\\) (left: 1, 2, 8)\n",
"12. \\(8 \\div 2 = 4\\) (left: 1, 1, 4)\n",
"\n",
"These are the possible next steps using basic arithmetic operations. Note that division operations only result in integer outcomes.\n"
"These are some of the possible operations you can perform with the given numbers.\n"
],
"values": [
0.001,
@ -41,236 +40,141 @@
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0.0,
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"select_new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n"
]
},
{
"step": 1,
"x": "1 1 2 8",
"ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n"
],
"new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\nLet's explore the possible next steps for the input numbers 1, 1, 2, and 8 by performing arithmetic operations between any two numbers and replacing them with the result. Here are the possible operations and resulting sets of numbers:\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n1. **Addition:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 + 1 = 2\\) (remaining: 2, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 + 2 = 3\\) (remaining: 1, 3, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 + 8 = 9\\) (remaining: 1, 2, 9)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 + 2 = 3\\) (remaining: 1, 3, 8) *(repeated operation)*\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 + 8 = 9\\) (remaining: 1, 2, 9) *(repeated operation)*\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(2 + 8 = 10\\) (remaining: 1, 1, 10)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n2. **Subtraction:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 - 1 = 0\\) (remaining: 0, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 - 2 = -1\\) (remaining: 1, -1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 - 8 = -7\\) (remaining: 1, 2, -7)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(2 - 1 = 1\\) (remaining: 1, 1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(8 - 1 = 7\\) (remaining: 1, 2, 7)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(8 - 2 = 6\\) (remaining: 1, 1, 6)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n3. **Multiplication:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 \\times 1 = 1\\) (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 \\times 2 = 2\\) (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 \\times 8 = 8\\) (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(2 \\times 1 = 2\\) (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(2 \\times 8 = 16\\) (remaining: 1, 1, 16)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n4. **Division:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(1 \\div 1 = 1\\) (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(2 \\div 1 = 2\\) (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(8 \\div 1 = 8\\) (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\(8 \\div 2 = 4\\) (remaining: 1, 1, 4)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\nThese are all the possible next steps by performing arithmetic operations on the input numbers 1, 1, 2, and 8.\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 2 = 3 (left: 1 3 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 8 = 9 (left: 1 2 9) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 2 = 3 (left: 1 3 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 8 = 9 (left: 1 2 9) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n2 + 8 = 10 (left: 1 1 10) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 - 1 = 0 (left: 0 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n2 - 1 = 1 (left: 1 1 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n8 - 1 = 7 (left: 1 2 7) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n2 - 1 = 1 (left: 1 1 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n8 - 1 = 7 (left: 1 2 7) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n8 - 2 = 6 (left: 1 1 6) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 * 1 = 1 (left: 1 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 * 2 = 2 (left: 1 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 * 8 = 8 (left: 1 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n2 * 8 = 16 (left: 1 1 16) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 / 1 = 1 (left: 1 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n2 / 1 = 2 (left: 1 1 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n8 / 1 = 8 (left: 1 2 8) \n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n8 / 2 = 4 (left: 1 1 4)\n"
],
"values": [
0.0,
0.001,
0.001,
0.001,
0.0,
0.001,
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"select_new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n"
]
},
{
"step": 2,
"x": "1 1 2 8",
"ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n"
],
"new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n1. **Addition:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 + 1 = 2 (remaining: 2, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 + 2 = 3 (remaining: 1, 3, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 + 8 = 9 (remaining: 1, 2, 9)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 2 + 8 = 10 (remaining: 1, 1, 10)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n2. **Subtraction:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 - 1 = 0 (remaining: 0, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 - 2 = -1 (remaining: 1, -1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 - 8 = -7 (remaining: 1, 2, -7)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 2 - 1 = 1 (remaining: 1, 1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 8 - 1 = 7 (remaining: 1, 2, 7)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 8 - 2 = 6 (remaining: 1, 1, 6)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n3. **Multiplication:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 * 1 = 1 (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 * 2 = 2 (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 * 8 = 8 (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 2 * 8 = 16 (remaining: 1, 1, 16)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n4. **Division:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 1 / 1 = 1 (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 2 / 1 = 2 (remaining: 1, 1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 8 / 1 = 8 (remaining: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n - 8 / 2 = 4 (remaining: 1, 1, 4)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nThese are all the possible operations and resulting sets of numbers that can be achieved from the given input numbers 1, 1, 2, and 8. Note that unnecessary operations that result in no change (e.g., multiplying by 1 or dividing by 1) can be omitted if desired.\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n1. \\(2 + 2 = 4\\) (left: 4 8)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n2. \\(2 \\times 2 = 4\\) (left: 4 8)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n3. \\(8 - 2 = 6\\) (left: 2 6)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n4. \\(8 - 2 = 6\\) (left: 2 6)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n5. \\(8 / 2 = 4\\) (left: 2 4)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n6. \\(8 / 2 = 4\\) (left: 2 4)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \n\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nNote: Steps 1 and 2 result in the same state (4 8), and steps 3 and 4 result in the same state (2 6), as do steps 5 and 6, resulting in the state (2 4).\n"
],
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"select_new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n"
]
},
{
"step": 3,
"x": "1 1 2 8",
"ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n"
],
"new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n1. **Addition:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 + 1 = 2 \\) (left: 2, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 + 2 = 3 \\) (left: 1, 1, 3, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 + 8 = 9 \\) (left: 1, 1, 2, 9)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 2 + 8 = 10 \\) (left: 1, 1, 10)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n2. **Subtraction:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 - 1 = 0 \\) (left: 0, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 2 - 1 = 1 \\) (left: 1, 1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 8 - 1 = 7 \\) (left: 1, 2, 7)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 8 - 2 = 6 \\) (left: 1, 1, 6)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n3. **Multiplication:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 \\times 1 = 1 \\) (left: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 \\times 2 = 2 \\) (left: 1, 1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 \\times 8 = 8 \\) (left: 1, 1, 2)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 2 \\times 8 = 16 \\) (left: 1, 1, 16)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n4. **Division:**\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 1 \\div 1 = 1 \\) (left: 1, 2, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 2 \\div 1 = 2 \\) (left: 1, 1, 8)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 8 \\div 1 = 8 \\) (left: 1, 1, 2)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n - \\( 8 \\div 2 = 4 \\) (left: 1, 1, 4)\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\n",
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\nNote: Division is only performed when it results in an integer.\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\nFor the input `2 2 8`, here are the possible next steps:\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n1. \\(2 + 2 = 4\\) (left: 4 8)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n2. \\(2 \\times 2 = 4\\) (left: 4 8)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n3. \\(8 - 2 = 6\\) (left: 2 6)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n4. \\(8 - 2 = 6\\) (left: 2 6)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n5. \\(8 \\div 2 = 4\\) (left: 2 4)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n6. \\(8 \\div 2 = 4\\) (left: 2 4)\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n\n",
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\nNote: Since the two `2`s are indistinguishable in operations like addition and multiplication, the results are the same, but I've listed them to show all possible pairings.\n"
],
"values": [
0.001,
0.001,
0.001,
0.0,
0,
0,
0.0,
0,
0.001,
0,
0.0,
0.0,
0.0,
0,
0.001,
0.0,
0.0,
0,
0,
0,
0.001,
0.0,
0.0,
0,
0.0,
0.0,
0,
0
],
"select_new_ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n\n"
]
}
],
"idx": 10,
"ys": [
"To find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\nTo find the possible next steps for the input numbers 1, 1, 2, and 8, we can perform arithmetic operations between any two numbers and replace them with the result. Here are the possible operations and the resulting sets of numbers:\n"
"Here are some possible next steps with the numbers 1, 1, 2, and 8:\n1 + 1 = 2 (left: 2 2 8) \nHere are the possible next steps for the input `2 2 8`:\n\n"
],
"infos": [
{
@ -278,361 +182,10 @@
}
],
"usage_so_far": {
"completion_tokens": 5801,
"prompt_tokens": 22299,
"cost": 1.0170299999999999
}
},
{
"steps": [
{
"step": 0,
"x": "1 1 4 8",
"ys": [
""
],
"new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n",
"\n",
"1. **Addition:**\n",
" - 1 + 1 = 2 (left: 2, 4, 8)\n",
" - 1 + 4 = 5 (left: 1, 5, 8)\n",
" - 1 + 8 = 9 (left: 1, 4, 9)\n",
" - 4 + 8 = 12 (left: 1, 1, 12)\n",
"\n",
"2. **Subtraction:**\n",
" - 1 - 1 = 0 (left: 0, 4, 8)\n",
" - 4 - 1 = 3 (left: 1, 3, 8)\n",
" - 8 - 1 = 7 (left: 1, 4, 7)\n",
" - 4 - 8 = -4 (left: 1, 1, -4)\n",
" - 8 - 4 = 4 (left: 1, 1, 4)\n",
"\n",
"3. **Multiplication:**\n",
" - 1 * 1 = 1 (left: 1, 4, 8)\n",
" - 1 * 4 = 4 (left: 1, 4, 8)\n",
" - 1 * 8 = 8 (left: 1, 4, 8)\n",
" - 4 * 8 = 32 (left: 1, 1, 32)\n",
"\n",
"4. **Division:**\n",
" - 1 / 1 = 1 (left: 1, 4, 8)\n",
" - 4 / 1 = 4 (left: 1, 4, 8)\n",
" - 8 / 1 = 8 (left: 1, 4, 8)\n",
" - 8 / 4 = 2 (left: 1, 1, 2)\n",
" - 4 / 8 = 0.5 (left: 1, 1, 0.5)\n",
"\n",
"These are the possible operations that can be performed with the numbers 1, 1, 4, and 8.\n"
],
"values": [
0.001,
0.001,
0.001,
0.0,
0.0,
0.0,
0.0,
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0.001,
0.0,
0.0,
0.0,
0.0,
0.0,
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"select_new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n"
]
},
{
"step": 1,
"x": "1 1 4 8",
"ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n"
],
"new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n1. Add 1 and 1:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 + 1 = 2 (remaining numbers: 2, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n2. Add 1 and 4:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 + 4 = 5 (remaining numbers: 1, 5, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n3. Add 1 and 8:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 + 8 = 9 (remaining numbers: 1, 4, 9)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n4. Add 4 and 8:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 + 8 = 12 (remaining numbers: 1, 1, 12)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n5. Subtract 1 from 1:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 - 1 = 0 (remaining numbers: 0, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n6. Subtract 1 from 4:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 - 1 = 3 (remaining numbers: 1, 1, 3)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n7. Subtract 1 from 8:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 - 1 = 7 (remaining numbers: 1, 4, 7)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n8. Subtract 4 from 8:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 - 4 = 4 (remaining numbers: 1, 1, 4)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n9. Multiply 1 and 1:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 * 1 = 1 (remaining numbers: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n10. Multiply 1 and 4:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 * 4 = 4 (remaining numbers: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n11. Multiply 1 and 8:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 * 8 = 8 (remaining numbers: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n12. Multiply 4 and 8:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 * 8 = 32 (remaining numbers: 1, 1, 32)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n13. Divide 1 by 1:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 / 1 = 1 (remaining numbers: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n14. Divide 4 by 1:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 / 1 = 4 (remaining numbers: 1, 1, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n15. Divide 8 by 1:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 / 1 = 8 (remaining numbers: 1, 1, 4)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n16. Divide 8 by 4:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 / 4 = 2 (remaining numbers: 1, 1, 2)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nNote: Dividing 1 by 4 or 1 by 8 results in fractions, which are not included here as they may not be considered valid operations in some contexts.\n"
],
"values": [
0.001,
0.001,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001,
0.001,
0,
0.001
],
"select_new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n"
]
},
{
"step": 2,
"x": "1 1 4 8",
"ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n"
],
"new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n1. Addition:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 + 1 = 2 (left: 2, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 + 4 = 5 (left: 1, 1, 5, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 + 8 = 9 (left: 1, 1, 4, 9)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 + 8 = 12 (left: 1, 1, 12)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n2. Subtraction:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 - 1 = 0 (left: 0, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 - 1 = 3 (left: 1, 1, 3, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 - 1 = 7 (left: 1, 1, 4, 7)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 - 4 = 4 (left: 1, 1, 4, 4)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n3. Multiplication:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 * 1 = 1 (left: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 * 4 = 4 (left: 1, 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 1 * 8 = 8 (left: 1, 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 * 8 = 32 (left: 1, 1, 32)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n4. Division (only when the result is an integer):\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 / 1 = 8 (left: 1, 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 4 / 1 = 4 (left: 1, 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n - 8 / 4 = 2 (left: 1, 1, 2)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nThese are the possible next steps you can take using arithmetic operations on the sequence 1, 1, 4, 8.\n"
],
"values": [
0.001,
0.001,
0.001,
0.0,
0.0,
0.0,
0.0,
0,
0.001,
0.0,
0,
0.0,
0.0,
0,
0.001,
0.0,
0.0,
0.0,
0.0,
0,
0.001,
0.0,
0.0,
0,
0,
0.001
],
"select_new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n"
]
},
{
"step": 3,
"x": "1 1 4 8",
"ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n"
],
"new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8 using arithmetic operations (addition, subtraction, multiplication, and division), we need to consider each pair of numbers and apply the operations. Here are the possible next steps:\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n1. **Addition:**\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 + 1 = 2\\) (left: 2, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 + 4 = 5\\) (left: 1, 5, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 + 8 = 9\\) (left: 1, 4, 9)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 + 4 = 5\\) (left: 1, 5, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 + 8 = 9\\) (left: 1, 4, 9)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(4 + 8 = 12\\) (left: 1, 1, 12)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n2. **Subtraction:**\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 - 1 = 0\\) (left: 0, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 - 4 = -3\\) (left: 1, -3, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 - 8 = -7\\) (left: 1, 4, -7)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(4 - 1 = 3\\) (left: 1, 3, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(8 - 1 = 7\\) (left: 1, 4, 7)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(4 - 8 = -4\\) (left: 1, 1, -4)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(8 - 4 = 4\\) (left: 1, 1, 4)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n3. **Multiplication:**\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 \\times 1 = 1\\) (left: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 \\times 4 = 4\\) (left: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 \\times 8 = 8\\) (left: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(4 \\times 8 = 32\\) (left: 1, 1, 32)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n4. **Division:**\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(1 / 1 = 1\\) (left: 1, 4, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(4 / 1 = 4\\) (left: 1, 1, 8)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(8 / 1 = 8\\) (left: 1, 4, 1)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(4 / 8 = 0.5\\) (left: 1, 1, 0.5)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n - \\(8 / 4 = 2\\) (left: 1, 1, 2)\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n\n",
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\nThese are the immediate next steps available by pairing and applying operations to the numbers in the sequence 1, 1, 4, 8.\n"
],
"values": [
0,
0.001,
0.001,
0.0,
0.0,
0.0,
0,
0,
0.0,
0,
0.001,
0.0,
0,
0,
0.0,
0.0,
0.0,
0.0,
0,
0.001,
0.0,
0.0,
0.0,
0.0,
0,
0.001,
0.0,
0.0,
0,
0.0,
0,
0,
0
],
"select_new_ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n\n"
]
}
],
"idx": 11,
"ys": [
"To determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, you can perform arithmetic operations (addition, subtraction, multiplication, and division) between any two numbers. After performing an operation, the two numbers are replaced with the result of the operation, reducing the total count of numbers by one. Here are the possible next steps:\nTo determine the possible next steps with the sequence 1, 1, 4, 8, we can perform arithmetic operations (addition, subtraction, multiplication, and division) on any two numbers in the sequence. Here are the possible next steps:\n\n"
],
"infos": [
{
"r": 0
}
],
"usage_so_far": {
"completion_tokens": 12723,
"prompt_tokens": 45580,
"cost": 2.13078
}
"completion_tokens": 5961,
"prompt_tokens": 12152,
"cost": 0.7222199999999999
},
"total_runtime": 95.51803588000075
}
]

13
run.py
View File

@ -22,22 +22,17 @@ def run(args):
for i in range(args.task_start_index, args.task_end_index):
# solve
start_timer = time.perf_counter()
if args.naive_run:
ys, info = naive_solve(args, task, i)
else:
ys, info = solve(args, task, i)
# print("HEY IM IN RUN AND I SOLVED")
# print("THIS IS THE Y")
# print(ys)
# print("ThIS IS THE INFO")
# print(info)
# exit()
runtime = time.perf_counter()-start_timer
print(runtime)
# log
infos = [task.test_output(i, y) for y in ys]
info.update({'idx': i, 'ys': ys, 'infos': infos, 'usage_so_far': gpt_usage(args.backend)})
info.update({'idx': i, 'ys': ys, 'infos': infos, 'usage_so_far': gpt_usage(args.backend), 'total_runtime': runtime})
logs.append(info)
with open(file, 'w') as f:
json.dump(logs, f, indent=4)

24
src/tot/methods/bfs.py
View File

@ -14,15 +14,10 @@ def get_value(task, x, y, n_evaluate_sample, cache_value=True):
return value
def get_values(task, x, ys, n_evaluate_sample, cache_value=True):
# ys = list(set(ys))
# print(ys)
values = []
local_value_cache = {}
for y in ys: # each partial output
# print("THIS IS THE Y")
# print(y)
# print("*****")
if y in local_value_cache: # avoid duplicate candidates
value = 0
else:
@ -40,10 +35,8 @@ def get_votes(task, x, ys, n_evaluate_sample):
def get_proposals(task, x, y):
propose_prompt = task.propose_prompt_wrap(x, y)
# print("prompt proposed -- now running inference...")
proposals = inference_model(propose_prompt, n=1, stop=None)[0].split('\n')
# print(proposals)
# print("*****=======")
return [y + _ + '\n' for _ in proposals]
def get_samples(task, x, y, n_generate_sample, prompt_sample, stop):
@ -72,26 +65,15 @@ def solve(args, task, idx, to_print=True):
# elif args.method_generate == 'propose':
else:
new_ys = [get_proposals(task, x, y) for y in ys]
# print("NEW YS BEFORE ITERTOOl")
# print(new_ys)
# print("**********")
new_ys = list(itertools.chain(*new_ys))
ids = list(range(len(new_ys)))
# print("THESE ARE THE NEW YS")
# print(new_ys)
# print("**********")
print("Starting eval!!")
# evaluation
if args.method_evaluate == 'vote':
values = get_votes(task, x, new_ys, args.n_evaluate_sample)
elif args.method_evaluate == 'value':
values = get_values(task, x, new_ys, args.n_evaluate_sample)
# print("THESE ARE THE VALUES")
# print(values)
# print("********")
print("Starting selection!!")
# selection
if args.method_select == 'sample':
ps = np.array(values) / sum(values)

7
src/tot/models.py
View File

@ -25,8 +25,7 @@ def inference_model(prompt, model="gpt-4o", temperature=0.7, max_tokens=1000, n=
'''
Driver function for model inference.
'''
# print("made it into driver.")
if model == "llama_3.2" and vllm:
if model == "llama_3.2" and vllm: #will change this later
return llama_32(prompt, quant, vllm, temperature, max_tokens, n, stop)
else:
messages = [{"role": "system", "content": "You are a helpful assistant."},
@ -56,7 +55,7 @@ def hf_model(model, input_tokens, temperature=0.7, max_tokens=1000, n=1, stop=No
outputs = []
while n > 0:
cnt = min(n, 20) #never generate more than 20 outputs per same input
cnt = min(n, 20)
n -= cnt
outputs = model.generate(**input_tokens, temperature=temperature, max_new_tokens=max_tokens, num_return_sequences=cnt) #might add stopping criteria depending on heuristics experimentation
#need to take a look at the specific output format once i get access to the gated repo
@ -79,7 +78,7 @@ def chatgpt(messages, model="gpt-4", temperature=0.7, max_tokens=1000, n=1, stop
res = client.chat.completions.create(model=model, messages=messages, temperature=temperature, n=cnt, stop=stop)
res_answer = res.choices[0].message.content
# print(res_answer)
outputs.extend([res_answer]) #will add in support for generate sample > 1 hyperparam later
outputs.extend([res_answer]) #will add in support for n > 1 hyperparam later
# log completion tokens
completion_tokens += res.usage.completion_tokens

4
src/tot/tasks/game24.py
View File

@ -74,10 +74,6 @@ class Game24Task(Task):
@staticmethod
def value_prompt_wrap(x: str, y: str) -> str:
# print("IM IN VALUE PROMPT WRAP")
# print(y)
# print(x)
# print("******")
last_line = y.strip().split('\n')[-1]
if 'left: ' not in last_line: # last step
ans = last_line.lower().replace('answer: ', '')