
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
Find all monic nonconstant polynomials  $P$  with integer coefficients for which there exist positive integers  $a$  and  $m$  such that for all positive integers  $n\equiv a\pmod m$ ,  $P(n)$  is nonzero and   $$ 2022\cdot\frac{(n+1)^{n+1} - n^n}{P(n)} $$   is an integer. 

*Jaedon Whyte, Luke Robitaille, and Pitchayut Saengrungkongka*

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