logs/step_116-b_0.log

{
  "log_file": [
    "step_116-b_0.log",
    "step_116-b_1.log",
    "step_116-b_2.log",
    "step_116-b_3.log",
    "step_116-b_4.log",
    "step_116-b_5.log"
  ],
  "query": [
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nHow many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nHow many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$\n\n### Response:\nSEARCH[\"number\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/counting_and_probability/269.json\n\n#### Similar Question\nWhat is the 39th number in the row of Pascal's triangle that has 41 numbers?\n\n#### User Answer\nThe row 1, 1 has 2 numbers.  The row 1, 2, 1 has 3 numbers.  The row 1, 3, 3, 1 has 4 numbers.  Each time we go down one row, we have one more number in the list.  So, the row that starts 1, $k$ has $k+1$ numbers (namely, the numbers $\\binom{k}{0}, \\binom{k}{1}, \\binom{k}{2}, \\ldots, \\binom{k}{k}$.)  So the row with 41 numbers starts $\\binom{40}{0}, \\binom{40}{1}, \\binom{40}{2}, \\ldots$.  The 39th number has two numbers after it, and it is the same as the number in the row with only two numbers before it (that is, the 39th number is the same as the 3rd).  So, the 39th number is $\\binom{40}{2} = \\frac{40\\cdot 39}{2\\cdot 1} = 780$.\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nHow many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nHow many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$\n\n### Response:\nSEARCH[\"number\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/counting_and_probability/543.json\n\n#### Similar Question\nHow many subsets of the set $\\{1,2,3,4,5\\}$ contain the number 5?\n\n#### User Answer\nSolution 1: For each of 1, 2, 3, and 4, we can either choose to include the number in the set, or choose to exclude it.  We therefore have 2 choices for each of these 4 numbers, which gives us a total of $2^4 = 16$ different subsets we can form.\n\nSolution 2: We can have either 5 by itself, 5 with one other number from the four, 5 with two other numbers, 5 with three other numbers, or 5 with all four other numbers. The number of ways to form a subset with 5 by itself is $\\binom{4}{0}$. The number of ways to form a subset with 5 and one other number is $\\binom{4}{1}$. Similarly, the number of ways to form a subset with 5 and two other numbers is $\\binom{4}{2}$, with three other numbers is $\\binom{4}{3}$, and with all four other numbers is $\\binom{4}{4}$. Thus, our answer is $1 + 4 + 6 + 4 + 1 = 16$.\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nHow many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nHow many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$\n\n### Response:\nSEARCH[\"number\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/counting_and_probability/235.json\n\n#### Similar Question\nWhat is the second number in the row of Pascal's triangle that has 43 numbers?\n\n#### User Answer\nThe row 1, 1 has 2 numbers.  The row 1, 2, 1 has 3 numbers.  The row 1, 3, 3, 1 has 4 numbers.  Each time we go down one row, we have one more number in the list.  So, the row that starts 1, $k$ has $k+1$ numbers (namely, the numbers $\\binom{k}{0}, \\binom{k}{1}, \\binom{k}{2}, \\ldots, \\binom{k}{k}$.)  So the row with 43 numbers starts 1, $42$.\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"number\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nNote that the numbers increase by 3 each time.  Hence we increase by 3 a total of $\\frac{32 - (-4)}{3} = 12$ times.  But then there must be $12 + 1 = \\boxed{13}$ numbers, since we need to also include the first number on the list.</s>",
    " SEARCH[\"number\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nNotice that the numbers increase by 3 each time.  We can use the formula for the number of terms in an arithmetic sequence to find the number of terms in the sequence:  \\begin{align*}\n\\frac{32 - (-4)}{3} + 1 &= 9 + 1\\\\\n13 + 1 &= \\boxed{14}\n\\end{align*}</s>",
    " SEARCH[\"number\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nAdd 3 to each member of the list to get the next member.  We obtain $-1$, $2$, $5$, $8$, $11$, $14$, $17$, $20$, $23$, $26$, $29$, $32$, $35$, $38$, $41$, $44$, $47$, $50$, $53$, $56$, $59$, $62$, $65$, $68$, $71$, $74$, $77$, $80$, $83$, $86$, $89$, $92$, $95$, $98$, $101$, $104$, $107$, $110$, $113$, $116$, $119$, $122$, $125$, $128$, $131$, $134$, $137$, $140$, $143$, $146$, $149$, $152$, $155$, $158$, $161$, $164$, $167$, $170$, $173$, $176$, $179$, $182$, $185$, $188$, $191$, $194$, $197$, $200$, $203$, $206$, $209$, $212$, $215$, $218$, $221$, $224$, $227$, $230$, $233$, $236$, $239$, $242$, $245$, $248$, $251$, $254$, $257$, $260$, $263$, $266$, $269$, $272$, $275$, $278$, $281$, $284$, $287$, $290$, $293$, $296$, $299$.  There are $\\boxed{198}$ numbers.</s>"
  ],
  "src_path": [
    "train/prealgebra/648.json",
    "train/prealgebra/648.json",
    "train/prealgebra/648.json",
    "train/prealgebra/648.json",
    "train/prealgebra/648.json",
    "train/prealgebra/648.json"
  ],
  "input": [
    "How many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$",
    "How many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$",
    "How many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$",
    "How many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$",
    "How many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$",
    "How many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$"
  ],
  "output": [
    "Note that the numbers increase by 3 each time.  Hence we increase by 3 a total of $\\frac{32 - (-4)}{3} = 12$ times.  But then there must be $12 + 1 = \\boxed{13}$ numbers, since we need to also include the first number on the list.",
    "Note that the numbers increase by 3 each time.  Hence we increase by 3 a total of $\\frac{32 - (-4)}{3} = 12$ times.  But then there must be $12 + 1 = \\boxed{13}$ numbers, since we need to also include the first number on the list.",
    "Note that the numbers increase by 3 each time.  Hence we increase by 3 a total of $\\frac{32 - (-4)}{3} = 12$ times.  But then there must be $12 + 1 = \\boxed{13}$ numbers, since we need to also include the first number on the list.",
    "Note that the numbers increase by 3 each time.  Hence we increase by 3 a total of $\\frac{32 - (-4)}{3} = 12$ times.  But then there must be $12 + 1 = \\boxed{13}$ numbers, since we need to also include the first number on the list.",
    "Note that the numbers increase by 3 each time.  Hence we increase by 3 a total of $\\frac{32 - (-4)}{3} = 12$ times.  But then there must be $12 + 1 = \\boxed{13}$ numbers, since we need to also include the first number on the list.",
    "Note that the numbers increase by 3 each time.  Hence we increase by 3 a total of $\\frac{32 - (-4)}{3} = 12$ times.  But then there must be $12 + 1 = \\boxed{13}$ numbers, since we need to also include the first number on the list."
  ],
  "rewards": [
    1.0,
    1.0,
    0.0,
    0.0,
    0.0,
    0.0
  ]
}