Diagnosing Failure

Read all of the following:

• ./problem_bank/$0/problem.md — the original problem statement.
• If $1 ≤ 4: read all plans (plan_1.md … plan_$1.md) and solutions (solution_1.py … solution_$1.py).
• If $1 > 4: read plan_1.md (original approach), plus the last 3 plans and solutions (plan_{$1-2}.md … plan_$1.md and solution_{$1-2}.py … solution_$1.py). This keeps context manageable while retaining the original approach and recent history.

Reading the history avoids repeating past mistakes and reveals patterns across attempts.

Step 1.5: Cross-trial pattern analysis

Skip this step entirely when $1 < 3 (i.e., this is the first or second trial reflection). Only activate from trial 3 onward.

Perform the following four checks using the full history gathered in Step 1:

A. Verdict pattern classification

Look at the verdict sequence across all trials and classify the pattern:

B. Algorithm continuity check

Has the fundamental algorithm stayed the same across trials? The following count as "same algorithm":

• Encoding changes (tuple → int, dict → array)
• Data structure swaps (list → deque, dict → defaultdict)
• Micro-optimizations (inlining, local variable caching, reducing constant factors)
• I/O changes (input method, output buffering)

A different algorithm means a different time-complexity class or a fundamentally different problem decomposition (e.g., DP → greedy, brute-force → divide-and-conquer, BFS → binary search).

C. Feasibility sanity check

Estimate actual resource usage for the current algorithm against the problem's constraints:

• Time: ~10^7 operations/sec for CPython, ~10^8 for PyPy. Multiply the algorithm's Big-O by the actual constant factors (inner loop operations, dictionary lookups, etc.).
• Memory: 28 bytes/int, 56 bytes/tuple, 50–100 bytes/dict entry, 56 bytes/list overhead + 8 bytes/pointer per element.

If the estimate exceeds 50% of either the time or memory limit, flag the algorithm as infeasible.

D. Stuck verdict

Declare STUCK if ANY of the following hold:

1. Pattern is Plateaued or Oscillating.
2. The fundamental algorithm has been unchanged for 3+ consecutive trials without the pattern being Progressing.
3. The feasibility check flags the current algorithm as infeasible.

If the pattern is Progressing, do NOT declare STUCK — the current approach is improving and should be continued.

Record the verdict (STUCK or NOT STUCK) and carry it forward to Step 2 and Step 3.

Step 2: Diagnose the failure

The verdict is $2. Analyze the failed solution against the problem statement. Focus your diagnosis based on the verdict:

Clearly state:

1. What went wrong — the specific bug or bottleneck.
2. Why — root cause tied back to the problem constraints or edge cases.
3. What the previous plan missed — which assumption or step was flawed.

Stuck-mode diagnosis

Only applies when Step 1.5 declared STUCK. Skip this subsection otherwise.

When STUCK is declared, the standard diagnosis above is insufficient. Additionally:

1. Prohibit further optimization of the current algorithm. No more encoding changes, data structure swaps, constant-factor reductions, or inlining. These have been tried and have failed.
2. State the fundamental limitation explicitly with concrete numbers (e.g., "A sliding-window DP over N=10^5 states with K=10^3 transitions requires 10^8 operations and 800 MB — both exceed limits").
3. Brainstorm 2+ fundamentally different algorithms — each must have a different complexity class or problem decomposition. For each candidate:
   • Describe the approach in 1–2 sentences.
   • Run the feasibility sanity check (Step 1.5C) against it.
   • State whether it passes or fails feasibility.
4. Select the most promising alternative that passes the feasibility check. If no candidate passes feasibility in Python/PyPy, explicitly recommend the user consider a different language.

Step 3: Produce a revised plan

Write an updated plan that fixes the diagnosed issue. The revised plan must include:

1. Reflection summary — verdict, what failed, and why (from Step 2).
2. Problem analysis — corrected understanding if the previous one was wrong; otherwise keep it.
3. Key observations — add any new observations; mark corrected ones.
4. Algorithm choice — keep or replace. If replacing, justify why the new approach avoids the previous failure. If Step 1.5 declared STUCK, you MUST choose a different algorithm. Justify why it avoids the previous failure mode and passes the feasibility check.
5. Step-by-step approach — updated implementation steps.
6. Edge cases — add any newly identified edge cases.
7. Complexity analysis — verify the revised approach fits within constraints.

[NOTE] DO NOT USE import sys. Instead use input() for reading input and print() for output.

[NOTE] DO NOT USE import io. Do NOT USE import os.

[NOTE] DO NOT import external packages like numpy, pandas, etc.

Step 4: Save the revised plan

Determine the output trial number:

• If $3 is provided, use {output_trial} = $3.
• Otherwise, use {output_trial} = $1 + 1.

Save the revised plan:

• Path: ./problem_bank/$0/plan_{output_trial}.md

If {output_trial} equals $1 (same trial — revising in place), overwrite the existing plan file.

After saving, the user should run /writing-solution $0 {output_trial} to implement the revised plan.