
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
Given a permutation \(\sigma\) of \(\{1, 2, \ldots, 2013\}\), let \(f(\sigma)\) be the number of fixed points of \(\sigma\) — that is, the number of \(k \in \{1, 2, \ldots, 2013\}\) such that \(\sigma(k) = k\). If \(S\) is the set of all possible permutations \(\sigma\), compute
\[ \sum_{\sigma \in S} f(\sigma)^{4}. \] 

(Here, a permutation \(\sigma\) is a bijective mapping from \(\{1, 2, \ldots, 2013\}\) to \(\{1, 2, \ldots, 2013\}\).)

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