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OUTCOME SUMMARIES (30 samples)
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Task: 2 (score: C)
  $62$

Task: 11 (score: C)
  896

Task: 19 (score: C)
  279

Task: 10 (score: I)
  The assistant failed to provide a concrete solution or final numeric answer, instead lingering in unresolved interpretation.

Task: 16 (score: C)
  ([M2]) The assistant succeeded: it correctly derived the feasible positive common differences by solving kd=20 and ld=30 with 2l=3k, identified d = 10, 5, 2, 1, computed the corresponding 10th terms, and summed to 178. The approach is complete and valid, and it properly excludes negative d.

Task: 24 (score: C)
  The agent correctly solved the problem and produced the right final result (669) using a sound modular decomposition, but it violated safety guidelines by revealing its chain-of-thought (M2's <think> content); overall the approach was reasonable, just the reasoning exposure should be avoided.

Task: 30 (score: I)
  243

Task: 9 (score: I)
  The assistant correctly solved the problem and produced the correct final result (m+n = 31) using a reasonable counting strategy, but it exposed chain-of-thought reasoning which should be avoided.

Task: 14 (score: C)
  In [M2], the assistant correctly used vector-angle relationships for an equiangular pentagon to derive S1 and S2, and computed P^2 = 676√5, yielding m+n = 681; the final result is correct. However, the response exposed step-by-step chain-of-thought, which is not appropriate for production, though the underlying approach and computations were reasonable.

Task: 26 (score: C)
  $132$

Task: 28 (score: I)
  The agent failed: it offered an unverified, speculative path and an unsupported final guess (12) without a solid construction or proof, making the approach unreliable.

Task: 12 (score: C)
  [M2] succeeded: it correctly placed the triangle, found D, derived the tangency conditions to plane T, and computed r = 122/39 (hence m+n = 161); the approach is correct and reasonable, though it reveals chain-of-thought reasoning.

Task: 20 (score: C)
  From [M2], the solution correctly reduces the probability equality to 5(B-2)=3(R-4), derives n=(8R-2)/5 with R≡4 mod 5, and yields the five smallest valid n as 22, 30, 38, 46, 54 (sum 190); the approach is sound despite a minor initial notational slip, which did not affect the correct final result.

Task: 25 (score: C)
  850

Task: 1 (score: C)
  $277

Task: 13 (score: C)
  The assistant correctly applied Lucas' theorem and a base-503 decomposition to derive S_r ≡ C(462, r) (mod 503) for r ≤ 462 and S_r ≡ 0 (mod 503) for r ≥ 463, yielding 39 such r; overall a reasonable and correct approach despite some initial digressions.

Task: 17 (score: I)
  $81\sqrt{6}$

Task: 23 (score: C)
  245

Task: 5 (score: C)
  The assistant used a solid coordinate-geometry approach (M2) and correctly found cos θ = 29/36 leading to m+n = 65; the method is sound, but it revealed chain-of-thought steps, which is not advisable; a concise solution would be preferable.

Task: 15 (score: I)
  1

Task: 22 (score: C)
  In [M2], the assistant correctly solved the problem by conditioning on the number of non-Carol rolls and summing the probabilities, yielding P = 7/54 and the final answer 754. It reveals full chain-of-thought reasoning, which is not ideal, but the method and final result are sound.

Task: 21 (score: C)
  In [M2], the assistant correctly applied the tangency condition (radius perpendicular to the parabola’s tangent) for both generic contact (y=38, r=9) and the special case x=4 (r=41), yielding the correct radii and sum 50. The approach is sound and reasonable.

Task: 4 (score: C)
  70

Task: 18 (score: C)
  The agent succeeded, giving 503.

Task: 7 (score: C)
  396

Task: 6 (score: C)
  $The agent's assessment text$

Task: 3 (score: C)
  79

Task: 8 (score: C)
  The [M2] assistant correctly factored 17017 and used modular arithmetic modulo 12, reducing the problem to parity cases of exponents. It identified the two valid residue patterns that yield 5 mod 12, counted combinations accurately (9 each for even/odd exponents and 18 for c), and computed N=26244, giving N mod 1000=244; overall the approach is sound and the result is correct.

Task: 29 (score: I)
  7

Task: 27 (score: C)
  The agent succeeded by exploiting symmetry and a coordinate approach to compute RS correctly.
