
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 \( S = \{1, 2, \ldots, 2021\} \), 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 
\[
\sum_{f \in \mathcal{F}} \left| T_{f} \right|
\]
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}
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