
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 number  $p$  such that there exists an integer-coefficient polynomial  $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$  that has  $p-1$  consecutive positive integer roots and  $p^2\mid f(i)f(-i)$ , where  $i$  is the imaginary unit.

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