
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For a polynomial \( P \) and a positive integer \( n \), define \( P_{n} \) as the number of positive integer pairs \( (a, b) \) such that \( a < b \leq n \) and \( |P(a)| - |P(b)| \) is divisible by \( n \).

Determine all polynomials \( P \) with integer coefficients such that for all positive integers \( n \), \( P_{n} \leq 2021 \).

Present the answer in LaTex format: \boxed{Your answer}
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