
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 the probability such that when a polynomial in  $\mathbb Z_{2027}[x]$  having degree at most  $2026$  is chosen uniformly at random,
      $$ x^{2027}-x | P^{k}(x) - x \iff 2021 | k  $$      (note that  $2027$  is prime).
    
    Here  $P^k(x)$  denotes  $P$  composed with itself  $k$  times.

*Proposed by Grant Yu*

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