
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]


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A polynomial  $P$  with real coefficients is called *great,* if for some integer  $a>1$  and for all integers  $x$ , there exists an integer  $z$  such that  $aP(x)=P(z)$ . Find all *great* polynomials.

*Proposed by A. Golovanov*

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