
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  $\alpha$  be a real number. Determine all polynomials  $P$  with real coefficients such that   $$ P(2x+\alpha)\leq (x^{20}+x^{19})P(x) $$   holds for all real numbers  $x$ .

*Proposed by Walther Janous, Austria*

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