
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 prime numbers \( p \) for which there exists a polynomial with integer coefficients
$$
f(x) = x^{p-1} + a_{p-2} x^{p-2} + \cdots + a_{1} x + a_{0},
$$

such that \( f(x) \) has \( p-1 \) consecutive positive integer roots, and \( p^{2} \) divides \( f(\mathrm{i}) f(-\mathrm{i}) \), where \( \mathrm{i} \) is the imaginary unit.

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