
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 \( f^{(i)}(x) = f\left(f^{(i-1)}(x)\right) \). What is the expected value of \( n \)?

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