
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
Suppose that  $P(x)$  is a monic cubic polynomial with integer roots, and suppose that  $\frac{P(a)}{a}$  is an integer for exactly  $6$  integer values of  $a$ . Suppose furthermore that exactly one of the distinct numbers  $\frac{P(1) + P(-1)}{2}$  and  $\frac{P(1) - P(-1)}{2}$  is a perfect square. Given that  $P(0) > 0$ , find the second-smallest possible value of  $P(0).$ *Proposed by Andrew Wu*

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