
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 an integer \( m \), let \( p(m) \) denote the largest prime factor of \( m \). For convenience, define \( p(\pm 1) = 1 \) and \( p(0) = \infty \). Find all integer-coefficient polynomials \( f \) such that the sequence \( \left\{p\left(f\left(n^{2}\right)\right) - 2n\right\}_{n \geqslant 0} \) is bounded above (in particular, this requires \( f\left(n^{2}\right) \neq 0 \) for any \( n \geqslant 0 \)).

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