
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 polynomials  $P$  with integer coefficients such that    $$ s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|). $$   for all  $x,y\in \mathbb{N}$ .
Note:  $s(x)$  denotes the sum of digits of  $x$ .

*Proposed by Şevket Onur YILMAZ*

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