logs/step_119-b_0.log

{
  "log_file": [
    "step_119-b_0.log",
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    "step_119-b_3.log",
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    "step_119-b_5.log"
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  "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:\nExpand the following expression: $3(8x^2-2x+1)$.\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:\nExpand the following expression: $3(8x^2-2x+1)$.\n\n### Response:\nSEARCH[\"3(8x^2-2x+1)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/233.json\n\n#### Similar Question\nFind the leading coefficient in the polynomial $-3(x^4 - x^3 + x) + 7(x^4 + 2) - 4(2x^4 + 2x^2 + 1)$ after it is simplified.\n\n#### User Answer\nThe leading coefficient is the coefficient of the term with the highest power of $x$, which in this case is $x^4$.  The coefficient of $x^4$ in $-3(x^4 - x^3 + x) + 7(x^4 + 2) - 4(2x^4 + 2x^2 + 1)$ is $-3 + 7 - 4 \\cdot 2 = -4$.\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:\nExpand the following expression: $3(8x^2-2x+1)$.\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:\nExpand the following expression: $3(8x^2-2x+1)$.\n\n### Response:\nSEARCH[\"3(8x^2-2x+1)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1605.json\n\n#### Similar Question\nLet $a,$ $b,$ $c$ be three distinct positive real numbers such that $a,$ $b,$ $c$ form a geometric sequence, and\n\\[\\log_c a, \\ \\log_b c, \\ \\log_a b\\]form an arithmetic sequence.  Find the common difference of the arithmetic sequence.\n\n#### User Answer\nSince $a,$ $b,$ $c$ form a geometric sequence, $b = \\sqrt{ac}.$  Then the three logarithms become\n\\[\\log_c a, \\ \\log_{\\sqrt{ac}} c, \\ \\log_a \\sqrt{ac}.\\]Let $x = \\log_c a.$  Then by the change-of-base formula,\n\\[\\log_{\\sqrt{ac}} c = \\frac{\\log_c c}{\\log_c \\sqrt{ac}} = \\frac{1}{\\frac{1}{2} \\log_c ac} = \\frac{2}{\\log_c a + \\log_c c} = \\frac{2}{x + 1},\\]and\n\\[\\log_a \\sqrt{ac} = \\frac{1}{2} \\log_a ac = \\frac{\\log_c ac}{2 \\log_c a} = \\frac{\\log_c a + \\log_c c}{2 \\log_c a} = \\frac{x + 1}{2x}.\\]Let $d$ be the common difference, so\n\\[d = \\frac{2}{x + 1} - x = \\frac{x + 1}{2x} - \\frac{2}{x + 1}.\\]Then\n\\[4x - 2x^2 (x + 1) = (x + 1)^2 - 4x,\\]which simplifies to $2x^3 + 3x^2 - 6x + 1 = 0.$  This factors as $(x - 1)(2x^2 + 5x - 1) = 0.$\n\nIf $x = 1,$ then $\\log_c a = 1,$ so $a = c.$  But $a$ and $c$ are distinct, so $2x^2 + 5x - 1 = 0,$ so $x^2 = \\frac{1 - 5x}{2}.$ Then\n\\[d = \\frac{2}{x + 1} - x = \\frac{2 - x^2 - x}{x + 1} = \\frac{2 - \\frac{1 - 5x}{2} - x}{x + 1} = \\frac{3x + 3}{2(x + 1)} = \\frac{3}{2}.\\]\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:\nExpand the following expression: $3(8x^2-2x+1)$.\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:\nExpand the following expression: $3(8x^2-2x+1)$.\n\n### Response:\nSEARCH[\"3(8x^2-2x+1)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/820.json\n\n#### Similar Question\nFind the roots of $6x^4+19x^3-51x^2+20x$. Enter your answer as a list of numbers separated by commas.\n\n#### User Answer\nSince the polynomial has no constant term, we can immediately factor out an $x$ from every term\n$$x(6x^3+19x^2-51x+20),$$and our first root $x=0$. Let $g(x) = 6x^3+19x^2-51x+20$. Then the remaining roots of our original polynomial are the roots of $g(x)$. By trying out simple values, we can see that $g(0) = 20 > 0$ and $g(1) = 6+19-51+20 = -6<0$. Thus, there must be a root of $g(x)$ between $0$ and $1$. From the Rational Root Theorem, we know that if $g(p/q) = 0$ then $p$ must divide $20$ and $q$ must divide $6$.\n\nChecking rational numbers of the form $p/q$, where $p$ divides $20$ and $q$ divides $6$, and $p/q$ is between $0$ and $1$, we find that\n$$\\begin{aligned} g\\left(\\frac{1}{2}\\right) &= 6\\cdot\\frac{1}{8}+19\\cdot\\frac{1}{4}-51\\cdot\\frac{1}{2}+20 = 0.\n\\end{aligned}$$This means that $2x - 1$ is a factor of $g(x).$ Dividing by $2x-1$ gives us $g(x) = (2x-1)(3x^2+11x-20)$.\n\nThe quadratic $3x^2+11x-20$ factors as $(3x-4)(x+5),$ so our last two roots are $4/3$ and $-5$.\n\nThus, the roots of $6x^4+19x^3-51x^2+20x$ are $0, \\frac{1}{2}, \\frac{4}{3}, -5$.\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"3(8x^2-2x+1)\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe apply the distributive property to get\\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2 - 3\\cdot 2x + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}.\n\\end{align*}</s>",
    " SEARCH[\"3(8x^2-2x+1)\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe have $3(8x^2-2x+1) = 24x^2 - 6x + 3 = 12x^2 - 3x + 3$.</s>",
    " SEARCH[\"3(8x^2-2x+1)\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe use the distributive property to expand.  \\begin{align*}\n3(8x^2-2x+1) &= 3(8x^2) + 3(-2x) + 3(1) \\\\\n&= \\boxed{24x^2 - 6x + 3}\n\\end{align*}</s>"
  ],
  "src_path": [
    "train/algebra/1688.json",
    "train/algebra/1688.json",
    "train/algebra/1688.json",
    "train/algebra/1688.json",
    "train/algebra/1688.json",
    "train/algebra/1688.json"
  ],
  "input": [
    "Expand the following expression: $3(8x^2-2x+1)$.",
    "Expand the following expression: $3(8x^2-2x+1)$.",
    "Expand the following expression: $3(8x^2-2x+1)$.",
    "Expand the following expression: $3(8x^2-2x+1)$.",
    "Expand the following expression: $3(8x^2-2x+1)$.",
    "Expand the following expression: $3(8x^2-2x+1)$."
  ],
  "output": [
    "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}",
    "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}",
    "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}",
    "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}",
    "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}",
    "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}"
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