
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  $S=\{1, 2, 3, \dots 2021\}$  and  $f:S \to S$  be a function such that  $f^{(n)}(n)=n$  for each  $n \in S$ .
Find all possible values for  $f(2021)$ .
(Here,  $f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))$ .)

*Proposed by Viktor Simjanoski*

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