span review: browser push

span_review/a845b086.json

@@ -816,6 +816,1358 @@
    "answer_correct": true,
    "grader_matches": true,
    "span_gates": []
+  },
+  "111:57": {
+   "status": "needs_review",
+   "gold_span": null,
+   "gold_text": null,
+   "ratified_by": null,
+   "ratified_at": null,
+   "reviewer": null,
+   "notes": "flagged for context review",
+   "row_id": "A0087"
+  },
+  "103:14": {
+   "status": "ratified",
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+   "gold_text": "Thus, the minimum value of \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0079",
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+   ],
+   "gold_text": "Thus, the minimum value of \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0079",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
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+  "106:12": {
+   "status": "needs_review",
+   "gold_span": null,
+   "gold_text": null,
+   "ratified_by": null,
+   "ratified_at": null,
+   "reviewer": null,
+   "notes": "flagged for context review",
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+  },
+  "106:46": {
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+   "gold_text": "The final answer is \\( \\boxed{ 1 } \\)<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0044",
+   "answer_correct": true,
+   "grader_matches": true,
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+  "110:8": {
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+   ],
+   "gold_text": "The final answer is \\( \\boxed{ 1 } \\)<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0044",
+   "answer_correct": true,
+   "grader_matches": true,
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+  "111:26": {
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+   ],
+   "gold_text": "The final answer is \\( \\boxed{ 1 } \\)<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0044",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
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+  "109:9": {
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+   ],
+   "gold_text": "The final answer is: \\(\\boxed{1}\\)<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0047",
+   "answer_correct": true,
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+  "103:3": {
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+   ],
+   "gold_text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0058",
+   "answer_correct": true,
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+  "111:18": {
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+   ],
+   "gold_text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0058",
+   "answer_correct": true,
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+  "111:10": {
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+   ],
+   "gold_text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0058",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
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+  "111:11": {
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+   ],
+   "gold_text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0058",
+   "answer_correct": true,
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+  "112:41": {
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+   ],
+   "gold_text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0058",
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+   ],
+   "gold_text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0058",
+   "answer_correct": true,
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+  "113:22": {
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+   ],
+   "gold_text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:42+00:00",
+   "reviewer": null,
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0070",
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+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0070",
+   "answer_correct": true,
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
+   "grader_matches": true,
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "113:16": {
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
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+  "113:56": {
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
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+  "113:23": {
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+   ],
+   "gold_text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0071",
+   "answer_correct": true,
+   "grader_matches": true,
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+   "gold_text": "So, the answer is 1.",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0017",
+   "answer_correct": true,
+   "grader_matches": true,
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+  "106:47": {
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+   "gold_text": "So, the answer is 1.",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0017",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "110:6": {
+   "status": "ratified",
+   "gold_span": [
+    2055,
+    11,
+    279,
+    4226,
+    374,
+    220,
+    16,
+    382
+   ],
+   "gold_text": "So, the answer is 1.",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0017",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "111:33": {
+   "status": "ratified",
+   "gold_span": [
+    576,
+    4226,
+    374,
+    220,
+    16,
+    382
+   ],
+   "gold_text": "The answer is 1.",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0033",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "106:62": {
+   "status": "ratified",
+   "gold_span": [
+    576,
+    4226,
+    374,
+    220,
+    16,
+    624
+   ],
+   "gold_text": "The answer is 1.",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0034",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "2:15": {
+   "status": "ratified",
+   "gold_span": [
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+    4226,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568
+   ],
+   "gold_text": "The answer is \\( \\boxed{1} \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0038",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "2:26": {
+   "status": "ratified",
+   "gold_span": [
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+    17767,
+    1124,
+    79075,
+    90,
+    220,
+    16,
+    335,
+    1124,
+    568
+   ],
+   "gold_text": "The final answer is \\( \\boxed{ 1 } \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0042",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "2:29": {
+   "status": "ratified",
+   "gold_span": [
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+    17767,
+    1124,
+    79075,
+    90,
+    220,
+    16,
+    335,
+    1124,
+    568
+   ],
+   "gold_text": "The final answer is \\( \\boxed{ 1 } \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0042",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "2:30": {
+   "status": "ratified",
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+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568
+   ],
+   "gold_text": "Equality holds when \\(A = B = C = \\frac{\\pi}{3}\\), so the minimum value of the expression is \\(\\boxed{1}\\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:43+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0002",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "106:42": {
+   "status": "ratified",
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+    90,
+    16,
+    92,
+    1124,
+    568
+   ],
+   "gold_text": "Equality occurs when \\( x = y = z = \\frac{1}{3} \\), so the minimum value of \\( x + y + z \\) is \\( \\boxed{1} \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:44+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0004",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "106:17": {
+   "status": "ratified",
+   "gold_span": [
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+    79075,
+    90,
+    16,
+    92,
+    1124,
+    3593
+   ],
+   "gold_text": "The minimum value is achieved when \\( A = B = C = \\frac{\\pi}{3} \\), so the answer is \\( \\boxed{1} \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:44+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0051",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "2:31": {
+   "status": "ratified",
+   "gold_span": [
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+    374,
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+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568
+   ],
+   "gold_text": "The minimum value is achieved when \\( A = B = C = \\frac{\\pi}{3} \\), so the answer is \\(\\boxed{1}\\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:44+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0052",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "112:15": {
+   "status": "ratified",
+   "gold_span": [
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+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568
+   ],
+   "gold_text": "Equality occurs when the triangle is equilateral, so the minimum value is \\( \\boxed{1} \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:44+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0005",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "2:25": {
+   "status": "ratified",
+   "gold_span": [
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+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568
+   ],
+   "gold_text": "Since we have found a case where the expression is equal to 1, the minimum value is \\(\\boxed{1}\\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:44+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0013",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "2:9": {
+   "status": "ratified",
+   "gold_span": [
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+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568
+   ],
+   "gold_text": "So, the minimum value is \\( \\boxed{1} \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:44+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0023",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
+  },
+  "113:19": {
+   "status": "ratified",
+   "gold_span": [
+    576,
+    4226,
+    374,
+    17767,
+    220,
+    16,
+    1124,
+    3593
+   ],
+   "gold_text": "The answer is \\( 1 \\).",
+   "ratified_by": "manual",
+   "ratified_at": "2026-07-07T05:56:44+00:00",
+   "reviewer": null,
+   "notes": null,
+   "row_id": "A0036",
+   "answer_correct": true,
+   "grader_matches": true,
+   "span_gates": []
   }
  },
  "candidates": [
@@ -2399,8 +3751,8 @@
    ],
    "text": "The answer is \\( \\boxed{1} \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -3416,8 +4768,8 @@
    ],
    "text": "The final answer is \\( \\boxed{ 1 } \\)<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -6011,8 +7363,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -6260,8 +7612,8 @@
    ],
    "text": "The final answer is \\( \\boxed{ 1 } \\)<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -6970,8 +8322,8 @@
    ],
    "text": "Equality holds when \\(A = B = C = \\frac{\\pi}{3}\\), so the minimum value of the expression is \\(\\boxed{1}\\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -7643,8 +8995,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -7818,8 +9170,8 @@
    ],
    "text": "The minimum value is achieved when \\( A = B = C = \\frac{\\pi}{3} \\), so the answer is \\( \\boxed{1} \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -8189,8 +9541,8 @@
    ],
    "text": "Equality occurs when the triangle is equilateral, so the minimum value is \\( \\boxed{1} \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -9681,8 +11033,8 @@
    ],
    "text": "Therefore, the minimum value of the expression \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is: \\[ \\boxed{1}. \\]<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -10776,8 +12128,8 @@
    ],
    "text": "The final answer is \\( \\boxed{ 1 } \\)<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -11008,426 +12360,426 @@
     60,
     151643
    ],
-   "text": "Therefore, the minimum value of the expression \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is: \\[ \\boxed{1}. \\]<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "106:59",
-   "source": "model:span_v2_strict@latest",
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-   "text": "Therefore, the minimum value of the given expression is \\( 3 \\sqrt[3]{\\left(\\tan\\left(\\frac{A}{2}\\right) \\tan\\left(\\frac{B}{2}\\right) \\tan\\left(\\frac{C}{2}\\right)\\right)^2} = 3 \\sqrt[3]{\\left(\\left(\\frac{1}{3}\\right)^{\\frac{3}{2}}\\right)^2} = 3 \\sqrt[3]{\\frac{1}{9}} = \\boxed{3} \\).",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2465:0",
-   "source": "model:span_v2_strict@latest",
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-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\(\\boxed{1}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2377:1",
-   "source": "model:span_v2_strict@latest",
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-    90,
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-    11035,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression \\(\\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right)\\) is \\(\\boxed{1}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2431:11",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
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-    897,
-    315,
-    279,
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-    1124,
-    11520,
-    79075,
-    90,
-    16,
-    11035,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression is \\(\\boxed{1}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2456:3",
-   "source": "model:span_v2_strict@latest",
-   "span": [
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-    374,
-    1124,
-    11520,
-    79075,
-    90,
-    18,
-    11035,
-    4292
-   ],
-   "text": "The answer is \\(\\boxed{3}\\).",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "106:12",
-   "source": "model:span_v2_strict@latest",
-   "span": [
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-    90,
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-   ],
-   "text": "Thus, the minimum value of the expression \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is: \\[ \\boxed{1}. \\]<|endoftext|>",
+   "text": "Therefore, the minimum value of the expression \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is: \\[ \\boxed{1}. \\]<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "106:59",
+   "source": "model:span_v2_strict@latest",
+   "span": [
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+    284,
+    1124,
+    79075,
+    90,
+    18,
+    92,
+    1124,
+    568
+   ],
+   "text": "Therefore, the minimum value of the given expression is \\( 3 \\sqrt[3]{\\left(\\tan\\left(\\frac{A}{2}\\right) \\tan\\left(\\frac{B}{2}\\right) \\tan\\left(\\frac{C}{2}\\right)\\right)^2} = 3 \\sqrt[3]{\\left(\\left(\\frac{1}{3}\\right)^{\\frac{3}{2}}\\right)^2} = 3 \\sqrt[3]{\\frac{1}{9}} = \\boxed{3} \\).",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2465:0",
+   "source": "model:span_v2_strict@latest",
+   "span": [
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+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\(\\boxed{1}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2377:1",
+   "source": "model:span_v2_strict@latest",
+   "span": [
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+    10699,
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+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression \\(\\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right)\\) is \\(\\boxed{1}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2431:11",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression is \\(\\boxed{1}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2456:3",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    576,
+    4226,
+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    18,
+    11035,
+    4292
+   ],
+   "text": "The answer is \\(\\boxed{3}\\).",
    "gates": [],
    "label": null,
    "label_source": null,
    "added_at": "2026-07-05T21:54:23+00:00"
   },
+  {
+   "rollout": "106:12",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    44500,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
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+    17767,
+    1124,
+    52591,
+    61,
+    17,
+    59,
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+    78045,
+    1124,
+    79075,
+    90,
+    16,
+    7810,
+    1124,
+    60,
+    151643
+   ],
+   "text": "Thus, the minimum value of the expression \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is: \\[ \\boxed{1}. \\]<|endoftext|>",
+   "gates": [],
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
   {
    "rollout": "113:17",
    "source": "model:span_v2_strict@latest",
@@ -13443,8 +14795,8 @@
    ],
    "text": "Equality occurs when \\( x = y = z = \\frac{1}{3} \\), so the minimum value of \\( x + y + z \\) is \\( \\boxed{1} \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -14094,8 +15446,8 @@
    ],
    "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -14995,8 +16347,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -15967,8 +17319,8 @@
    ],
    "text": "Since we have found a case where the expression is equal to 1, the minimum value is \\(\\boxed{1}\\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -20379,8 +21731,8 @@
    ],
    "text": "So, the minimum value is \\( \\boxed{1} \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -21611,8 +22963,8 @@
    ],
    "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -22072,433 +23424,433 @@
     16,
     92,
     1124,
-    568
-   ],
-   "text": "So, the minimum value of \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\( \\boxed{1} \\).",
-   "gates": [],
-   "label": "correct",
-   "label_source": "autograde_vs_gold",
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "8060:34",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    576,
-    1590,
-    4226,
-    374,
-    1124,
-    11520,
-    79075,
-    90,
-    16,
-    11035,
-    568,
-    151643
-   ],
-   "text": "The final answer is \\(\\boxed{1}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "103:13",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    576,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    18,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "The minimum value of the expression is \\( \\boxed{3} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "113:20",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    16,
-    92,
-    1124,
-    568
-   ],
-   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "112:21",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    18,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression is \\( \\boxed{3} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "6643:17",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    16,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2464:4",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    1124,
-    26056,
-    37018,
-    90,
-    16,
-    481,
-    1124,
-    9407,
-    362,
-    15170,
-    16,
-    488,
-    1124,
-    9407,
-    362,
-    92,
-    488,
-    1124,
-    37018,
-    90,
-    16,
-    481,
-    1124,
-    9407,
-    425,
-    15170,
-    16,
-    488,
-    1124,
-    9407,
-    425,
-    92,
-    488,
-    1124,
-    37018,
-    90,
-    16,
-    481,
-    1124,
-    9407,
-    356,
-    15170,
-    16,
-    488,
-    1124,
-    9407,
-    356,
-    92,
-    1124,
-    709,
-    80,
-    220,
-    18,
-    1124,
-    26888,
-    58,
-    18,
-    15370,
-    59,
-    37018,
-    90,
-    16,
-    15170,
-    23,
-    3417,
-    284,
-    1124,
-    79075,
-    90,
-    18,
-    92,
-    7110
-   ],
-   "text": "Therefore, \\[\\frac{1 - \\cos A}{1 + \\cos A} + \\frac{1 - \\cos B}{1 + \\cos B} + \\frac{1 - \\cos C}{1 + \\cos C} \\geq 3 \\sqrt[3]{\\frac{1}{8}} = \\boxed{3}.\\",
-   "gates": [
-    "no_terminal",
-    "unclosed \\["
-   ],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2448:10",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    576,
-    1590,
-    4226,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    220,
-    18,
-    335,
-    1124,
-    568,
-    151643
-   ],
-   "text": "The final answer is \\( \\boxed{ 3 } \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "8060:41",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    54815,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    16,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2:27",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    576,
-    1590,
-    4226,
-    374,
-    25,
-    17767,
-    1124,
-    79075,
-    90,
-    220,
-    18,
-    335,
-    1124,
-    568
-   ],
-   "text": "The final answer is: \\( \\boxed{ 3 } \\).",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "106:22",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    8704,
-    279,
-    9334,
-    315,
-    264,
-    1931,
-    1372,
-    374,
-    2677,
-    2477,
-    60935,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    16,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "Since the square of a real number is always non-negative, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "112:0",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    14301,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    17767,
-    1124,
-    52591,
-    61,
-    17,
-    59,
-    2359,
-    11520,
-    37018,
-    90,
-    32,
-    15170,
-    17,
-    11035,
-    1291,
-    8,
-    488,
-    1124,
-    52591,
-    61,
-    17,
-    59,
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-    11520,
-    37018,
-    90,
-    33,
-    15170,
-    17,
-    11035,
-    1291,
-    8,
-    488,
-    1124,
-    52591,
-    61,
-    17,
-    59,
-    2359,
-    11520,
-    37018,
-    90,
-    34,
-    15170,
-    17,
-    11035,
-    1291,
-    8,
-    1124,
-    8,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    16,
-    92,
-    1124,
+    568
+   ],
+   "text": "So, the minimum value of \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\( \\boxed{1} \\).",
+   "gates": [],
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "8060:34",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    576,
+    1590,
+    4226,
+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
     568,
     151643
    ],
-   "text": "Thus, the minimum value of \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\( \\boxed{1} \\).<|endoftext|>",
+   "text": "The final answer is \\(\\boxed{1}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "103:13",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    576,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    18,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "The minimum value of the expression is \\( \\boxed{3} \\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "113:20",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568
+   ],
+   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).",
+   "gates": [],
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "112:21",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    18,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression is \\( \\boxed{3} \\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "6643:17",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2464:4",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    1124,
+    26056,
+    37018,
+    90,
+    16,
+    481,
+    1124,
+    9407,
+    362,
+    15170,
+    16,
+    488,
+    1124,
+    9407,
+    362,
+    92,
+    488,
+    1124,
+    37018,
+    90,
+    16,
+    481,
+    1124,
+    9407,
+    425,
+    15170,
+    16,
+    488,
+    1124,
+    9407,
+    425,
+    92,
+    488,
+    1124,
+    37018,
+    90,
+    16,
+    481,
+    1124,
+    9407,
+    356,
+    15170,
+    16,
+    488,
+    1124,
+    9407,
+    356,
+    92,
+    1124,
+    709,
+    80,
+    220,
+    18,
+    1124,
+    26888,
+    58,
+    18,
+    15370,
+    59,
+    37018,
+    90,
+    16,
+    15170,
+    23,
+    3417,
+    284,
+    1124,
+    79075,
+    90,
+    18,
+    92,
+    7110
+   ],
+   "text": "Therefore, \\[\\frac{1 - \\cos A}{1 + \\cos A} + \\frac{1 - \\cos B}{1 + \\cos B} + \\frac{1 - \\cos C}{1 + \\cos C} \\geq 3 \\sqrt[3]{\\frac{1}{8}} = \\boxed{3}.\\",
+   "gates": [
+    "no_terminal",
+    "unclosed \\["
+   ],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2448:10",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    576,
+    1590,
+    4226,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    220,
+    18,
+    335,
+    1124,
+    568,
+    151643
+   ],
+   "text": "The final answer is \\( \\boxed{ 3 } \\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "8060:41",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    54815,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2:27",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    576,
+    1590,
+    4226,
+    374,
+    25,
+    17767,
+    1124,
+    79075,
+    90,
+    220,
+    18,
+    335,
+    1124,
+    568
+   ],
+   "text": "The final answer is: \\( \\boxed{ 3 } \\).",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "106:22",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    8704,
+    279,
+    9334,
+    315,
+    264,
+    1931,
+    1372,
+    374,
+    2677,
+    2477,
+    60935,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "Since the square of a real number is always non-negative, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
    "label": null,
    "label_source": null,
    "added_at": "2026-07-05T21:54:23+00:00"
   },
+  {
+   "rollout": "112:0",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    14301,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    17767,
+    1124,
+    52591,
+    61,
+    17,
+    59,
+    2359,
+    11520,
+    37018,
+    90,
+    32,
+    15170,
+    17,
+    11035,
+    1291,
+    8,
+    488,
+    1124,
+    52591,
+    61,
+    17,
+    59,
+    2359,
+    11520,
+    37018,
+    90,
+    33,
+    15170,
+    17,
+    11035,
+    1291,
+    8,
+    488,
+    1124,
+    52591,
+    61,
+    17,
+    59,
+    2359,
+    11520,
+    37018,
+    90,
+    34,
+    15170,
+    17,
+    11035,
+    1291,
+    8,
+    1124,
+    8,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "Thus, the minimum value of \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\( \\boxed{1} \\).<|endoftext|>",
+   "gates": [],
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
   {
    "rollout": "2407:14",
    "source": "model:span_v2_strict@latest",
@@ -22705,8 +24057,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -22803,8 +24155,8 @@
    ],
    "text": "The final answer is: \\(\\boxed{1}\\)<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -22829,8 +24181,8 @@
    ],
    "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -23855,8 +25207,8 @@
    ],
    "text": "The final answer is \\( \\boxed{ 1 } \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -24220,8 +25572,8 @@
    ],
    "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -24470,8 +25822,8 @@
    ],
    "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -24777,8 +26129,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -25476,8 +26828,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -25510,257 +26862,257 @@
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
-   "rollout": "2:1",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    2055,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    18,
-    92,
-    1124,
-    568
-   ],
-   "text": "So, the minimum value of the expression is \\( \\boxed{3} \\).",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2454:3",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    2055,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    18,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "So, the minimum value of the expression is \\( \\boxed{3} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2447:14",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    2055,
-    279,
-    4226,
-    374,
-    1124,
-    11520,
-    79075,
-    35702,
-    37018,
-    90,
-    16,
-    15170,
-    18,
-    3417,
-    59,
-    568,
-    151643
-   ],
-   "text": "So the answer is \\(\\boxed{\\frac{1}{3}}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2412:6",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    1124,
-    11520,
-    79075,
-    90,
-    16,
-    11035,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression is \\(\\boxed{1}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "113:22",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    16,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "2367:3",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    1447,
-    59,
-    9640,
-    59,
-    79075,
-    90,
-    16,
-    27275,
-    59,
-    2533
-   ],
-   "text": "Therefore, the minimum value of the expression is: \\[ \\boxed{1}. \\]",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "112:54",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    374,
-    1124,
-    11520,
-    79075,
-    90,
-    18,
-    11035,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value is \\(\\boxed{3}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "8060:52",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    14301,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    1124,
-    11520,
-    79075,
-    90,
-    16,
-    11035,
-    568,
-    151643
-   ],
-   "text": "Thus, the minimum value of the expression is \\(\\boxed{1}\\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
-   "rollout": "113:23",
-   "source": "model:span_v2_strict@latest",
-   "span": [
-    15277,
-    11,
-    279,
-    8028,
-    897,
-    315,
-    279,
-    7493,
-    374,
-    17767,
-    1124,
-    79075,
-    90,
-    16,
-    92,
-    1124,
-    568,
-    151643
-   ],
-   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
-   "gates": [],
-   "label": null,
-   "label_source": null,
-   "added_at": "2026-07-05T21:54:23+00:00"
-  },
-  {
+   "rollout": "2:1",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    2055,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    18,
+    92,
+    1124,
+    568
+   ],
+   "text": "So, the minimum value of the expression is \\( \\boxed{3} \\).",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2454:3",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    2055,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    18,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "So, the minimum value of the expression is \\( \\boxed{3} \\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2447:14",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    2055,
+    279,
+    4226,
+    374,
+    1124,
+    11520,
+    79075,
+    35702,
+    37018,
+    90,
+    16,
+    15170,
+    18,
+    3417,
+    59,
+    568,
+    151643
+   ],
+   "text": "So the answer is \\(\\boxed{\\frac{1}{3}}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2412:6",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression is \\(\\boxed{1}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "113:22",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
+   "gates": [],
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "2367:3",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    1447,
+    59,
+    9640,
+    59,
+    79075,
+    90,
+    16,
+    27275,
+    59,
+    2533
+   ],
+   "text": "Therefore, the minimum value of the expression is: \\[ \\boxed{1}. \\]",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "112:54",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    18,
+    11035,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value is \\(\\boxed{3}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "8060:52",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    14301,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    1124,
+    11520,
+    79075,
+    90,
+    16,
+    11035,
+    568,
+    151643
+   ],
+   "text": "Thus, the minimum value of the expression is \\(\\boxed{1}\\).<|endoftext|>",
+   "gates": [],
+   "label": null,
+   "label_source": null,
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
+   "rollout": "113:23",
+   "source": "model:span_v2_strict@latest",
+   "span": [
+    15277,
+    11,
+    279,
+    8028,
+    897,
+    315,
+    279,
+    7493,
+    374,
+    17767,
+    1124,
+    79075,
+    90,
+    16,
+    92,
+    1124,
+    568,
+    151643
+   ],
+   "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
+   "gates": [],
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
+   "added_at": "2026-07-05T21:54:23+00:00"
+  },
+  {
    "rollout": "8060:31",
    "source": "model:span_v2_strict@latest",
    "span": [
@@ -25864,8 +27216,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -27068,8 +28420,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -27159,8 +28511,8 @@
    ],
    "text": "Therefore, the minimum value is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -28882,8 +30234,8 @@
    ],
    "text": "Therefore, the minimum value of the expression is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -29286,8 +30638,8 @@
    ],
    "text": "The minimum value is achieved when \\( A = B = C = \\frac{\\pi}{3} \\), so the answer is \\(\\boxed{1}\\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -29784,8 +31136,8 @@
    ],
    "text": "Thus, the minimum value of \\( \\tan^2\\left(\\frac{A}{2}\\right) + \\tan^2\\left(\\frac{B}{2}\\right) + \\tan^2\\left(\\frac{C}{2}\\right) \\) is \\( \\boxed{1} \\).<|endoftext|>",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -30380,8 +31732,8 @@
    ],
    "text": "The final answer is \\( \\boxed{ 1 } \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T21:54:23+00:00"
   },
   {
@@ -41963,8 +43315,8 @@
    ],
    "text": "So, the answer is 1.",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T22:03:36+00:00"
   },
   {
@@ -42715,8 +44067,8 @@
    ],
    "text": "So, the answer is 1.",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T22:03:36+00:00"
   },
   {
@@ -43783,8 +45135,8 @@
    ],
    "text": "The answer is 1.",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T22:03:36+00:00"
   },
   {
@@ -44703,8 +46055,8 @@
    ],
    "text": "The answer is 1.",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T22:03:36+00:00"
   },
   {
@@ -45220,8 +46572,8 @@
    ],
    "text": "So, the answer is 1.",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T22:03:36+00:00"
   },
   {
@@ -45680,8 +47032,8 @@
    ],
    "text": "The answer is \\( 1 \\).",
    "gates": [],
-   "label": null,
-   "label_source": null,
+   "label": "correct",
+   "label_source": "autograde_vs_gold",
    "added_at": "2026-07-05T22:03:36+00:00"
   },
   {