
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  $S = \{1, 2, \dots , 2021\}$ , and let  $\mathcal{F}$  denote the set of functions  $f : S \rightarrow S$ . For a function  $f \in \mathcal{F},$  let
\[T_f =\{f^{2021}(s) : s \in S\},\]
where  $f^{2021}(s)$  denotes  $f(f(\cdots(f(s))\cdots))$  with  $2021$  copies of  $f$ . Compute the remainder when
\[\sum_{f \in \mathcal{F}} |T_f|\]
is divided by the prime  $2017$ , where the sum is over all functions  $f$  in  $\mathcal{F}$ .

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