
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 integral \( m \), let \( p(m) \) be the greatest prime divisor of \( m \). By convention, we set \( p(\pm 1)=1 \) and \( p(0)=\infty \). Find all polynomials \( f \) with integer coefficients such that the sequence
\[
\left\{p\left(f\left(n^{2}\right)\right)- 2n\right\}_{n \geq 0}
\]
is bounded above. (In particular, this requires \( f\left(n^{2}\right) \neq 0 \) for \( n \geq 0 \).)

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