
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  $P(x)$  be the unique polynomial of degree at most  $2020$  satisfying  $P(k^2)=k$  for  $k=0,1,2,\dots,2020$ . Compute  $P(2021^2)$ .

*Proposed by Milan Haiman.*

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