
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 \( p \) and \( q \) be distinct prime numbers, and let \( n \geq 3 \) be a positive integer. Find all integers \( a \) such that the polynomial \( f(x) = x^n + ax^{n-1} + pq \) can be factored into a product of two non-constant polynomials with integer coefficients.

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