
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 us consider a polynomial  $P(x)$  with integers coefficients satisfying  $$ P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156. $$  What is the largest possible number of integers  $x$  satisfying  $$ P(P(x))=x^2? $$  

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