
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 polynomials  $f$  with integer coefficients satisfying the following condition: there exists a positive integer  $N$  such that  $p$  divides  $2(f(p)!)+1$  for every prime  $p>N$  for which  $f(p)$  is a positive integer.

*Note: A monic polynomial has a leading coefficient equal to 1.*

*(Greece - Panagiotis Lolas and Silouanos Brazitikos)*

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