
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
Given distinct prime numbers  $p$  and  $q$  and a natural number  $n \geq 3$ , find all  $a \in \mathbb{Z}$  such that the polynomial  $f(x) = x^n + ax^{n-1} + pq$  can be factored into 2 integral polynomials of degree at least 1.

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