Explanation: The model attempt is completely empty, indicating it failed to generate any response or ran out of tokens.


Explanation: The model attempt is completely empty, indicating it failed to generate a response or ran out of tokens.


Explanation: The model attempt is completely empty, indicating it failed to generate any response.


Explanation: The model attempt is completely empty, indicating it failed to generate any response or ran out of tokens before producing an answer.


Explanation: The model attempt is completely empty, providing no reasoning or final answer.


Explanation: The model produced no output, likely due to a timeout or generation failure.


Explanation: The model produced no output or ran out of tokens before providing an answer.


Explanation: The model attempt is completely empty, providing no reasoning or answer.


Explanation: The model incorrectly assumed the rectangle vertices must be vertices of the dodecagon, and then its response was cut off before reaching any final answer.


Explanation: The planner attempted to answer the problem directly without decomposing it into sub-queries, resulting in a completely incorrect guess.


Explanation: The planner failed to decompose the problem into sub-queries and instead directly guessed an incorrect answer of 1 without any meaningful mathematical reasoning.


Explanation: The planner failed to decompose the problem into sub-queries and instead attempted to answer directly, resulting in an arbitrary and completely incorrect guess.


Explanation: The planner bypassed the executor and attempted to answer the problem directly, resulting in a completely baseless guess (likely just 2024 + 1).


Explanation: The planner attempted to solve the problem directly without delegating any sub-tasks to the executor, resulting in a completely incorrect final answer.


Explanation: The planner attempted to solve the problem directly without delegating any sub-tasks to the executor, resulting in a hallucinated and completely incorrect expression.


Explanation: The planner attempted to answer the problem directly without delegating any sub-tasks to the executor, resulting in a completely incorrect guess instead of calculating the actual probability.


Explanation: The model incorrectly assumed that the vertices of the rectangles must be vertices of the dodecagon. It failed to realize that the sides of the rectangles can be formed by segments of intersecting diagonals, allowing for many more rectangles.


Explanation: The planner completely hallucinated a mathematical relationship, assuming the inradius is simply the product of the number of circles and their radius (N*r = 2024). It failed to set up the actual geometric equations relating the chain of circles to the angle of the triangle and the inradius.


Explanation: The planner attempted to solve the problem directly without delegating to the executor, resulting in an incorrect count of valid colorings and an incorrect final probability.


Explanation: The model was on the right track and correctly deduced the structure of the maximal configurations, but the response cut off before it could compute the final answer.


Explanation: The planner attempted to solve the problem directly without delegating any steps to the executor, resulting in a hallucinated and completely incorrect answer.


Explanation: The model assumed the vertices of the rectangles must be vertices of the dodecagon, calculating only the 15 such rectangles, and completely missed all rectangles formed by intersections of diagonals.


Explanation: The planner asked a single sub-query to analyze the function h(t), but then immediately guessed a final answer of 12 without actually setting up or solving for the intersections of the two equations.


12 in the numerator (103 instead of 115)

Explanation: The planner attempted to solve the problem directly but miscounted the number of valid colorings (likely when applying the Principle of Inclusion-Exclusion to the sets of colorings valid for each rotation), finding 103 instead of 115.


Explanation: The model assumed the vertices of the rectangle must be vertices of the dodecagon (yielding 6 choose 2 = 15 rectangles). It failed to account for rectangles whose vertices are intersections of perpendicular diagonals/sides.


Explanation: The planner assumed the rectangles must have their vertices at the vertices of the dodecagon, which only yields 15 rectangles. It ignored the problem statement and diagram showing that rectangles can be formed by the intersections of orthogonal diagonals.


Explanation: The planner correctly deduced that at least one variable must be 100 and found 201 solutions for each case, but failed to use the Principle of Inclusion-Exclusion to subtract the overcounted cases where multiple variables equal 100.


Explanation: The executor's response was cut off mid-sentence while evaluating the piecewise function, causing the planner to abort the problem-solving process and output a random guess.


Explanation: The planner hallucinated an irrational value for AP and then arbitrarily guessed that the answer should be 25/13, bypassing any rigorous geometric calculation like using Ptolemy's Theorem on the harmonic quadrilateral.


Explanation: The planner incorrectly assumed the rectangles must have their vertices on the dodecagon, missing that the sides of the rectangles just need to lie on the diagonals or sides (allowing vertices to be intersection points). This led to simply calculating 6 choose 2 instead of counting all valid rectangles.


Explanation: The planner incorrectly assumed the number of humps was simply 4 and 3 based on the coefficients of the trigonometric functions, completely ignoring the nested absolute value functions which multiply the frequency. It then incorrectly assumed the number of intersections is just the product of these humps.


Explanation: The planner attempted to answer directly but failed to produce any final answer.


Explanation: The planner attempted to solve the complex combinatorial problem directly without delegating to the executor, resulting in an incorrect count of valid chip configurations.


Explanation: The planner incorrectly assumed that the vertices of the rectangles must be vertices of the dodecagon. The problem actually asks for rectangles formed by the intersections of any sides or diagonals, which requires choosing pairs of parallel lines in perpendicular directions.


Explanation: The planner failed to include the total pool size (10 numbers) in its sub-query to the executor. This self-containment failure confused the executor, whose response was cut off, leading the planner to hallucinate a completely incorrect final answer.


Explanation: The planner assumed the rectangles must have their vertices on the dodecagon (inscribed rectangles), missing that the problem allows rectangles formed by the intersections of any sides or diagonals. This led to calculating 6 choose 2 = 15 instead of finding pairs of perpendicular chords.


Explanation: The planner failed to produce any final answer, likely due to running out of tokens or generating an empty response.


Explanation: The planner used an incorrect formula for the maximum real part of $c_1 z + c_2 / z$. Instead of expanding the expression into $A\cos\theta + B\sin\theta$ and finding the maximum via $\sqrt{A^2+B^2}$, it hallucinated a flawed trigonometric formula involving the sum of the arguments.


Explanation: The planner correctly deduced there are 201 solutions when a=100, but simply multiplied this by 3 to get 603. It failed to use inclusion-exclusion to subtract the overlaps, double-counting the triple (100, 100, 100).


Explanation: The planner completely failed to understand the geometric configuration, baselessly guessing a linear relationship r = (N+1)R instead of setting up the proper trigonometric equations for the chain of circles.


Explanation: The model attempt is completely empty, indicating it failed to generate any response or ran out of tokens.


Explanation: The model attempt is completely empty, indicating it failed to generate any response.


Explanation: The planner incorrectly assumed that rectangles ABCD and EFGH lie on the same side of the line containing D, E, C, F. This led to the wrong y-coordinates for H and G (17 instead of -17), resulting in the incorrect quadratic equation for x_E.


Explanation: The planner attempted to solve the problem directly without delegating, likely guessing a simple formula like 2^10 - 2 = 1022 which fails to account for the complex combinatorial constraints of the maximal grid placements.


Explanation: The planner assumed the rectangle EFGH extended in the same direction as ABCD (y > 0), leading to the power of a point equation x_E * x_F = 17. It missed the valid geometric case where EFGH extends in the opposite direction (y < 0), which yields x_E * x_F = 561 and leads to the correct integer solution.


Explanation: The planner misinterpreted the condition 'all blue vertices end up at red positions' as requiring the rotated blue vertices to exactly equal the red vertices (B+k = B^c). This caused it to miss all valid colorings where the number of blue vertices is less than 4.


Explanation: The planner failed to decompose the problem into manageable geometric steps, instead repeatedly asking the executor to solve the entire problem or vague questions. The executor failed to answer, leading to a completely guessed final answer.


Explanation: The planner incorrectly assumed that the vertices of the rectangles must be the vertices of the dodecagon. It completely missed that rectangles can be formed by the internal intersections of the diagonals, as shown in the problem's diagram.


Explanation: The model correctly set up the Shoelace formula and computed the sum of coefficients as 1, which would yield the correct area of 588, but the generation cut off before stating the final answer.


48 in the numerator (163 instead of 115)

Explanation: The planner incorrectly assumed that all subsets of size <= 4 satisfy the condition by confusing the number of pairs (6) with the number of directed differences (up to 12). This led to counting all 70 subsets of size 4 as valid, whereas only 22 actually satisfy the disjoint shift condition.


Explanation: The model attempt is completely empty, providing no reasoning or final answer.


Explanation: The model's response was cut off mid-sentence before it could finish its calculations or provide a final answer.


Explanation: The planner correctly identified that subsets of size > 4 are all 'Bad', but failed to realize that the problem asks for the probability of a 'Good' configuration, which means the total number of Good configurations is exactly the 115 found (93 + 22). The planner incorrectly calculated 70 - 24 = 46 for size 4, missing that there are 48 bad subsets of size 4, not 24.


Explanation: The planner incorrectly calculated the number of monotonic segments for the composed functions. It found that $S(x)$ has 4 segments and $H(u)$ has 4 segments, but failed to multiply them to get 16 segments for $F(x)$ (and similarly 24 for $G(y)$), leading to a massive undercounting of the intersections.


Explanation: The planner failed to set up a system of equations to solve for the two unknowns (n_1 and n_4) simultaneously. Instead, it incorrectly attempted to find n_1 using only the total population equation, forcing the executor to assume n_4=0.


Explanation: The planner assumed the rectangle EFGH was on the same side of the line as ABCD (y=17), missing the case where it is on the opposite side (y=-17). This led to an unsolvable non-integer coordinate for the circle's center.


Explanation: The model attempt is completely empty, indicating it failed to generate a response.


Explanation: The model attempt is completely empty, indicating it failed to generate any response or ran out of tokens.


Explanation: The model attempt is completely empty, indicating it failed to generate a response or ran out of tokens.


Explanation: The model attempt is completely empty, indicating it failed to generate any response.


Explanation: The model attempt is completely empty, indicating it failed to generate a response or ran out of tokens before producing anything.


Explanation: The planner bypassed all the required 3D geometry by baselessly guessing that the difference in the radii of the tangency circles is simply the diameter of the torus tube. It then instructed the executor to compute the final answer based on this fabricated assumption.


Explanation: The planner assumed rectangle EFGH was on the same side of line DC as ABCD (setting y=17 for H and G), missing the valid case where it lies on the opposite side (y=-17). This missed case led to the wrong constant term in the quadratic equation for the x-coordinates, yielding an irrational answer.


Explanation: The planner correctly used the executor to find the maximum squared diameter of the box (657/16), but failed to divide by 4 to find the squared radius. Furthermore, it failed to format the final answer as p+q.


Explanation: The model incorrectly calculated the number of ways as 2^10 - 2 = 1022 by independently assigning colors to rows and columns. It failed to account for the boundary cases where a color is entirely absent, which overcounts the configurations since column choices become meaningless when no rows have that color.


Explanation: The planner incorrectly assumed that the vertices of the rectangles must coincide with the vertices of the dodecagon. The problem only requires the sides of the rectangles to lie on the sides or diagonals of the dodecagon, allowing for intersections of diagonals to be the rectangle's vertices.


\begin{aligned}
&2B + 3C + D \equiv 1 \pmod 7 \\
&6A + 3C + D \equiv 5 \pmod 7 \\
&6A + 2B + D \equiv 4 \pmod 7 \\
&6A + 2B + 3C \equiv 6 \pmod 7
\end{aligned}

Explanation:
5.
$B \equiv 6 \pmod 7 \implies B=6$.
$C \equiv 2 \pmod 7 \implies C=2$ or $9$.
$D \equiv 4 \pmod 7 \implies D=4$.
So the greatest is $A=5, B=6, C=9, D=4 \implies 5694$.
Then $Q = 5$, $R = 694$.
$Q+R


Explanation: The model failed to produce any final answer, terminating without outputting a value.


Explanation: The planner misunderstood the geometric arrangement of the circles, assuming they were stacked along an angle bisector from vertex B to side AC. The correct configuration has the circles arranged in a chain along one side of the triangle (e.g., AC), with the first circle tangent to AB and the last tangent to BC.


Explanation: The planner attempted to answer directly but failed to produce any final answer.


Explanation: The planner attempted to solve the problem directly without decomposing it into sub-queries, leading to a completely incorrect probability and final answer.


Explanation: The planner failed to decompose the problem or provide a direct answer, resulting in an empty final output.


Explanation: The planner failed to decompose the problem into sub-tasks and instead directly output a guessed answer of 0 without performing the necessary geometric and algebraic steps.


fraction instead of integer, close magnitude

Explanation: The planner attempted to solve the problem directly without delegating to the executor, resulting in an incorrect fractional expected value instead of the correct integer 204.


Explanation: The planner failed to delegate any tasks or produce any reasoning, resulting in an empty final answer.


Explanation: The planner attempted to solve the problem directly without delegating to the executor, resulting in a flawed counting approach that yielded exactly 1/5 of the correct answer.


Explanation: The planner attempted to solve the problem directly without delegating to the executor, resulting in an incorrect count of the valid triples.


Explanation: The planner attempted to answer the problem directly without delegating to executors, but failed to produce any final answer.


Explanation: The planner attempted to solve the problem directly without decomposing it or delegating to the executor, resulting in a completely incorrect guess.


Explanation: The executor failed to properly solve the sub-query and just output 7 without any derivation, leading the planner to adopt this baseless answer.


Explanation: The planner hallucinated the final formula for the number of solutions as N=3^23 without doing any actual mathematical derivation or decomposition. It then just asked the executor to compute 3^23 mod 1000, leading to an incorrect final answer.


Explanation: The planner attempted to solve the problem directly without delegating any sub-queries, failing to correctly calculate the expected number of intersections between the random chords and the diameters.


Explanation: The planner attempted to solve the problem directly without decomposing it into sub-queries, resulting in an incorrect final answer.


Explanation: The planner incorrectly converted the two-digit base-b number 97_b into the quadratic polynomial 9b^2+7b+1 instead of the correct linear expression 9b+7.


Explanation: The planner incorrectly expanded the equation $|z-4-k|=|z-3i-k|$ to get the line $(2k+8)x - (2k+6)y - (2k+7) = 0$ instead of the correct line $8x - 6y - 8k - 7 = 0$. This algebra error led to a completely wrong quadratic equation for $k$.


Explanation: The planner attempted to answer the problem directly without decomposing it into sub-queries, resulting in a completely incorrect guess.


Explanation: The planner incorrectly assumed the total number of valid grids is simply 9!, completely misunderstanding the row and block constraints of the Sudoku-like grid. It solved the core problem incorrectly in its own thought process and only used the executor for basic arithmetic.


Explanation: The planner attempted to solve the problem directly without delegating any geometric calculations to the executor, resulting in a flawed mental calculation or guess.


Explanation: The planner asked a single sub-query to find the total number of partitions, but after receiving an incomplete response from the executor, it abruptly halted and hallucinated a final answer without attempting to solve the actual probability question.


Explanation: The model attempted to solve the problem directly without delegation and severely miscounted the valid pairings (likely finding 512 instead of the correct 1920 valid pairings), leading to an incorrect numerator.


Explanation: The model attempt is completely empty, indicating it failed to generate any response or ran out of tokens.


Explanation: The planner incorrectly identified the complex point k+3i as (0, 3+k) instead of (k, 3). This led it to derive a completely wrong tangency equation to send to the executor.


Explanation: The planner incorrectly assumed the probability of two random segments intersecting is 1/3 (yielding E[RR]=100), failing to account for the condition that each segment's endpoints must lie in different quadrants. This led to an incorrect expected number of intersections.


Explanation: The planner bypassed the executor and attempted to solve the problem directly, failing to correctly analyze the modular arithmetic conditions and arriving at a completely wrong count.


Explanation: The model's combinatorial reasoning and prime factorization steps are entirely correct, but the response was cut off before it could compute the final sum of p*a.


Explanation: The planner attempted to solve the problem directly without delegating to the executor, resulting in a severely flawed count of the values where the greedy algorithm fails. It vastly underestimated the number of failures (finding only 14 instead of 390).


Explanation: The planner answered directly without delegation, erroneously assuming the three values of k were simply the three given roots (18, 72, 98) and summing them to get 188.


Explanation: The planner correctly calculated the answer as 610 in its internal reasoning, but sent a context-free verification query to the executor. After the executor understandably failed to verify it, the planner inexplicably changed its final answer to 600.


Explanation: The planner provided a general formula of 2^g for the number of matchings, which fails for the boundary case m=12 where the cycle length is 2. A cycle of length 2 (a single edge) has only 1 perfect matching, not 2, causing the executor to add 4096 instead of 1.


Explanation: The planner incorrectly assumed the number of solutions is exactly $3^{18}/3^7 = 3^{11}$ based on a uniform distribution heuristic, completely ignoring the actual number-theoretic constraints and cases (such as when a, b, c are permutations of 0, 1, 2 mod 3).


Explanation: The planner correctly computed N = 2907 using the executor, but forgot to take the remainder when divided by 1000 for the final answer.


Explanation: The model correctly identified the valid roots and the extraneous root, but likely made a minor arithmetic error when summing the rational parts of the roots (e.g., adding an extra 1/34), leading to a numerator of 3 instead of 1 and a final sum of 261 instead of 259.


Explanation: The planner correctly noted that interior gaps must be >= 1, but then used the unconstrained Stars and Bars formula \binom{8+M}{M}, which assumes all gaps can be 0. The correct number of ways to distribute the empty seats is \binom{9}{M}.


Explanation: The planner attempted to solve the problem directly without decomposing it into sub-queries, likely using a flawed heuristic or incorrect algebraic symmetry assumption to find the values of k.


Explanation: The planner hallucinated the value of $N_2=45$ (the true value is 81) and then invented a completely incorrect formula $N_n = \frac{5}{9} 3^{2n}$ based on this hallucination.


Explanation: The planner asked the executor to compute the next term of a sequence without providing the recurrence relation or initial values. This caused the executor to hallucinate a response based on Fibonacci numbers.


Explanation: The executor's computation was cut off mid-sentence before it could finish calculating the area of triangle ABC. As a result, the planner lacked the necessary information and hallucinated a final answer of 1284 instead of the correct 588.


Explanation: The planner hallucinated the probability of two random chords intersecting as 8/9 instead of the correct 17/36. It then passed this incorrect formula to the executor to merely perform the arithmetic.


Explanation: The planner incorrectly set up the intersection equation as 36(y+4)(y+1)^2 = (3y^2 - y - 12)^2 instead of the correct 12(y+4)(y+1)^2 = (3y^2 - y - 12)^2. This hallucinated coefficient led to completely wrong quadratic factors and an incorrect final answer.


Explanation: The model bypassed the planner-executor process and simply guessed the answer by summing the three given roots (18 + 72 + 98 = 188), completely failing to analyze the conditions for the function's minima.


Explanation: The planner incorrectly assumed that the values of k would simply be the other three roots (18, 72, 98) to create a double root, and directly output their sum without verifying if this actually results in two equal global minima.


Explanation: The planner attempted to answer directly without delegating to the executor but failed to produce any final answer.


Explanation: The planner incorrectly calculated the area of the region as 507\sqrt{2} instead of 507\sqrt{3}, likely by using the wrong projection factor (\sqrt{2} instead of \sqrt{3}) for a region on the plane x+y+z=75. This resulted in b=2 instead of b=3.


Explanation: The planner attempted to solve the problem directly without delegating to the executor, failing to correctly identify the recurrence for the sum of the numerator and denominator and evaluate the modular arithmetic.


Explanation: The planner failed to decompose the problem or provide any answer, resulting in an empty final output.


Explanation: The planner failed to delegate any tasks or produce a final answer, resulting in an empty output.


Explanation: The planner completely failed to analyze the given inequalities to find the true vertices of the region. Instead, it merely calculated the area of the first-octant triangle and then hallucinated a final answer.


Explanation: The planner incorrectly assumed that the 2025th term of the sequence would exactly equal its limit of 1/2, bypassing any actual evaluation of the recurrence. It then directly instructed the executor to output 3.


Explanation: The planner incorrectly calculated arc FG as 48 degrees by confusing it with arc DG (the arc between the midpoint and altitude foot on the same side, which is 2(B-C) = 48). Arc FG actually measures 72 degrees, leading to an incorrect arithmetic query.


Explanation: The planner incorrectly assumed the rational product \prod \frac{k^2-1}{k^2-4} telescopes to just the last numerator over the first denominator (giving (63^2-1)/(4^2-4) = 992/3), which when multiplied by the log part (3) yields 992, leading to m+n=993.


Explanation: The planner failed to mathematically determine the values of k and instead made a baseless guess that the sum of the k values would simply be the sum of the other three roots (18+72+98=188).


Explanation: The planner omitted the crucial constraint BC = 38 when passing the problem to the executor. Without this length, the triangle's dimensions are not fixed and the area cannot be determined.


Explanation: The planner incorrectly calculated the expected number of chord-chord intersections. It assumed a probability of 13/24 without properly accounting for the conditional probability of intersection given that the endpoints of each chord must lie in different quadrants.


Explanation: The executor failed to complete the algebraic elimination and root-finding for the quartic equation, returning an empty response. The planner then hallucinated or defaulted to a final answer of 23, which is completely incorrect.


Explanation: The planner failed to include the initial value and the recurrence relation in its second sub-query. As a result, the executor hallucinated a standard cotangent recurrence and symbolic initial value, leading to a completely incorrect final answer.


Explanation: The planner correctly calculated the areas of triangle ABC and the side triangles ABK and ALC, but failed to subtract the area of the central equilateral triangle AKL (49\sqrt{3}) when finding the area of quadrilateral BKLC.


Explanation: The planner incorrectly assumed that the geometric median of a pentagon is at a vertex if its internal angle is >= 120 degrees, which is a property of triangles, not pentagons. It directly solved the problem using this false assumption instead of properly decomposing the minimization problem.


Explanation:
{error_category}: "conceptual_error",
  "error_stage": "direct_answer",
  "off_by": "500",
  "explanation": "The model attempted to solve the problem directly without decomposing it, likely making an


Explanation: The planner incorrectly calculated the probability of two random chords intersecting as 8/9. By properly analyzing the cases of which quadrants the endpoints fall into, the true probability of intersection is 23/36, leading to a vastly overestimated number of regions.


Explanation: The model answered directly without decomposing the problem, likely guessing or using a flawed heuristic to arrive at 288. The correct approach requires setting P(x) + Mx = (x^2 - Ax + B)^2 and solving for k, which yields k = 8, 32, and 200, summing to 240.


Explanation: The planner asked a trivial sub-query about the number of distinct values in a few cells and then abruptly output a final answer without actually calculating the number of valid grid configurations.


Explanation: The planner abruptly abandoned its casework for the different valuations of 3 and hallucinated a final answer of 781 without completing the necessary counting.


Explanation: The planner failed to decompose the complex geometric optimization problem, instead querying the executor to solve the entire problem in a single step. When the executor failed to provide a valid mathematical derivation, the planner resorted to guessing an incorrect final answer.


Explanation: The planner correctly deduced the formula for the number of matchings per step k (2^{#cycles}), but failed to include this formula in the final sub-query. As a result, the executor had to solve the problem from scratch and miscounted the total as 4208 instead of 113.


Explanation: The planner incorrectly assumed that the function achieving its minimum at two points means its derivative's numerator Q(x) has a multiple root. The correct condition is that the two local minima have the same value, which occurs when the numerator of f(x) - m is the square of a quadratic.


Explanation:
 respectively, standard convention usually implies this order. Let's a"
It got cut off.
Then Turn 2 planner state before:
FACTS:
1. $\triangle ABC$ has angles $A=84^\circ, B=60^\circ, C=36^\circ$.
2.


Explanation: The planner failed to find a valid geometric method to minimize the distance sum, instead baselessly guessing that vertex A was the optimal point X. This led to evaluating f(A) and incorrectly concluding the minimum was 40 + 20√3.


Explanation: The executor failed to return a response in Turn 2, causing the planner to hallucinate the roots and the sum of the y-coordinates. The planner completely fabricated the values a=203, b=5, c=185, d=34, leading to an incorrect final answer.


Explanation: The planner misread the problem statement, assuming there were 27 random segments in addition to the 2 diameters (total 29 lines), rather than 25 random segments (total 27 lines).


factor of 2 for arc HJ (12 instead of 24)

Explanation: The executor incorrectly computed the measure of minor arc HJ as 12 degrees instead of 24 degrees. It likely found the inscribed angle subtending the arc (which is 12 degrees) and failed to double it to get the actual arc measure.


Explanation: The executor abruptly stopped its calculation for AD and hallucinated a boxed answer of 47. The testing framework caught this as the final answer, prematurely terminating the run before the planner could actually solve the problem.


Explanation: The planner correctly set up the expression for I_mix as `2 * 25 * P(cross)` with `P(cross) = 2/3`, but incorrectly calculated `2 * 25 * 2/3` as `50/3` instead of `100/3`.


Explanation: The planner incorrectly analyzed the intersection points, missing the valid negative roots for the positive slope segments and completely dismissing all intersections on the negative slope segments, leading to a completely wrong summation formula.


Explanation: The planner failed to decompose the complex geometry problem, instead passing the entire configuration to the executor in a single query. Without breaking down the problem into manageable steps (e.g., identifying the nine-point circle, finding the locations of the altitude feet, and calculating individual arc lengths), the executor was unable to correctly determine the arc measures.


Explanation: The planner correctly identified the quartic equation for the critical points in Turn 1, but in Turn 3 inexplicably truncated it to a quadratic by dropping the constant term and dividing by x^2. This algebraic mistake led to formulating all subsequent sub-queries based on a completely incorrect equation for the minima.