
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 pairs of integers  $a,b$  for which there exists a polynomial  $P(x) \in \mathbb{Z}[X]$  such that product  $(x^2+ax+b)\cdot P(x)$  is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0  \] where each of  $c_0,c_1,\ldots,c_{n-1}$  is equal to  $1$  or  $-1$ .

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