
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  $S = \{1, 2, \ldots, 2016\}$ , and let  $f$  be a randomly chosen bijection from  $S$  to itself. Let  $n$  be the smallest
positive integer such that  $f^{(n)}(1) = 1$ , where  $f^{(i)}(x) = f(f^{(i-1)}(x))$ . What is the expected value of  $n$ ?


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