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

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