
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
Let  $f(x)=x^n + a_{n-1}x^{n-1} + ...+ a_0 $  be a polynomial of degree  $ n\ge 1 $  with  $ n$  (not necessarily distinct) integer roots. Assume that there exist distinct primes  $p_0,p_1,..,p_{n-1}$  such that  $a_i > 1$  is a power of  $p_i$ , for all  $ i=0,1,..,n-1$ . Find all possible values of  $ n$ .

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