
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  $P$  be a polynom of degree  $n \geq 5$  with integer coefficients given by  $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$  with  $a_i \in \mathbb{Z}$ ,  $a_n \neq 0$ .

Suppose that  $P$  has  $n$  different integer roots (elements of  $\mathbb{Z}$ ) :  $0,\alpha_2,\ldots,\alpha_n$ . Find all integers  $k \in \mathbb{Z}$  such that  $P(P(k))=0$ .

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