{
  "log_file": [
    "step_1-b_0.log",
    "step_1-b_1.log",
    "step_1-b_2.log",
    "step_1-b_3.log",
    "step_1-b_4.log",
    "step_1-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:\nFind the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 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:\nFind the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]\n\n### Response:\nSEARCH[\"y^2 = 27 - x^2 = 11\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1434.json\n\n#### Similar Question\nFind the distance between the foci of the ellipse\n\\[\\frac{x^2}{20} + \\frac{y^2}{4} = 7.\\]\n\n#### User Answer\nFirst, we divide both sides by 7, to get\n\\[\\frac{x^2}{140} + \\frac{y^2}{28} = 1.\\]Thus, $a^2 = 140$ and $b^2 = 28,$ so $c^2 = a^2 - b^2 = 140 - 28 = 112.$  Thus, $c = \\sqrt{112} = 4 \\sqrt{7},$ so the distance between the foci is $2c = 8 \\sqrt{7}.$\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:\nFind the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 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:\nFind the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]\n\n### Response:\nSEARCH[\"y^2 = 27 - x^2 = 11\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/152.json\n\n#### Similar Question\nIn a certain hyperbola, the center is at $(-2,0),$ one focus is at $(-2 + \\sqrt{34},0),$ and one vertex is at $(-5,0).$  The equation of this hyperbola can be written as\n\\[\\frac{(x - h)^2}{a^2} - \\frac{(y - k)^2}{b^2} = 1.\\]Find $h + k + a + b.$\n\n#### User Answer\nThe center of the hyperbola is $(h,k) = (-2,0).$  The distance between the center and one vertex is $a = 3,$ and the distance between the center and one focus is $c = \\sqrt{34}.$  Then $b^2 = c^2 - a^2 = 34 - 3^2 = 25,$ so $b = 5.$\n\nTherefore, $h + k + a + b = -2 + 0 + 3 + 5 = 6.$\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:\nFind the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 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:\nFind the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]\n\n### Response:\nSEARCH[\"y^2 = 27 - x^2 = 11\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/229.json\n\n#### Similar Question\nSuppose that $a$ and $b$ are nonzero integers such that two of the roots of\n\\[x^3 + ax^2 + bx + 9a\\]coincide, and all three roots are integers.  Find $|ab|.$\n\n#### User Answer\nLet the integer roots be $r,$ $r,$ and $s,$ so\n\\[x^3 + ax^2 + bx + 9a = (x - r)^2 (x - s).\\]Expanding and matching coefficients, we get\n\\begin{align*}\n2r + s  &= -a, \\\\\nr^2 + 2rs &= b, \\\\\nr^2 s &= -9a.\n\\end{align*}From the first and third equations, $r^2 s = 9(2r + s),$ so\n\\[s r^2 - 18r - 9s = 0.\\]As a quadratic in $r,$ the discriminant is\n\\[\\sqrt{18^2 - 4(s)(-9s)} = \\sqrt{324 + 36s^2} = 3 \\sqrt{s^2 + 9}.\\]Since $r$ and $s$ are integers, $s^2 + 9$ must be a perfect square.  Let $s^2 + 9 = d^2,$ where $d > 0.$  Then\n\\[(d + s)(d - s) = 9.\\]If $s = 0,$ then $a = 0,$ which is not allowed.  Otherwise, $d = \\pm 5$ and $s = \\pm 4.$  If $s = 4,$ then $r = 6,$ and $a = -16$ and $b = 84.$  If $s = -4,$ then $r = -6,$ and $a = 16$ and $b = 84.$  In either case,\n\\[|ab| = 16 \\cdot 84 = 1344.\\]\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"y^2 = 27 - x^2 = 11\"]</s>",
    " The result might be helpful, I will try using it to answer the question only if it is useful.\n\nLet $a$ and $b$ be the semi-major and semi-minor axes of the hyperbola, respectively. Then, we have $a^2 = 27$ and $b^2 = 11$. The distance between the vertices of the hyperbola is twice the distance between the center of the hyperbola and one of the vertices. Therefore, the distance between the vertices of the hyperbola is $2 \\left(\\frac{a^2 + b^2}{2}\\right) = 2 \\left(\\frac{27 + 11}{2}\\right) = 2 \\left(\\frac{38}{2}\\right) = \\boxed{38}$.</s>",
    " SEARCH[\"y^2 = 27 - x^2 = 11\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe read that $a^2 = 27$ and $b^2 = 11,$ so $c^2 = a^2 + b^2 = 40.$  Therefore, $c = 2 \\sqrt{10}.$  The distance between the vertices is $2c = \\boxed{4 \\sqrt{10}}.$. The answer is 4\\sqrt{10}</s>",
    " SEARCH[\"y^2 = 27 - x^2 = 11\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe center of the hyperbola is $(5,0).$  The vertices of the hyperbola are at $(5 \\pm 2 \\sqrt{6}, \\pm 2 \\sqrt{6} \\cdot 3).$  Taking the positive sign, we find that the distance between the vertices is\n\\[\\sqrt{(5 + 2 \\sqrt{6})^2 + (2 \\sqrt{6} \\cdot 3)^2} = \\boxed{6 \\sqrt{6}}.\\]</s>"
  ],
  "src_path": [
    "train/intermediate_algebra/197.json",
    "train/intermediate_algebra/197.json",
    "train/intermediate_algebra/197.json",
    "train/intermediate_algebra/197.json",
    "train/intermediate_algebra/197.json",
    "train/intermediate_algebra/197.json"
  ],
  "input": [
    "Find the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]",
    "Find the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]",
    "Find the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]",
    "Find the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]",
    "Find the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]",
    "Find the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\]"
  ],
  "output": [
    "We read that $a^2 = 27,$ so $a = \\sqrt{27} = 3 \\sqrt{3}.$  Therefore, the distance between the vertices is $2a = \\boxed{6 \\sqrt{3}}.$",
    "We read that $a^2 = 27,$ so $a = \\sqrt{27} = 3 \\sqrt{3}.$  Therefore, the distance between the vertices is $2a = \\boxed{6 \\sqrt{3}}.$",
    "We read that $a^2 = 27,$ so $a = \\sqrt{27} = 3 \\sqrt{3}.$  Therefore, the distance between the vertices is $2a = \\boxed{6 \\sqrt{3}}.$",
    "We read that $a^2 = 27,$ so $a = \\sqrt{27} = 3 \\sqrt{3}.$  Therefore, the distance between the vertices is $2a = \\boxed{6 \\sqrt{3}}.$",
    "We read that $a^2 = 27,$ so $a = \\sqrt{27} = 3 \\sqrt{3}.$  Therefore, the distance between the vertices is $2a = \\boxed{6 \\sqrt{3}}.$",
    "We read that $a^2 = 27,$ so $a = \\sqrt{27} = 3 \\sqrt{3}.$  Therefore, the distance between the vertices is $2a = \\boxed{6 \\sqrt{3}}.$"
  ],
  "rewards": [
    0.0,
    0.0,
    0.0,
    0.0,
    0.0,
    0.0
  ]
}