
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 \([n]\) denote the set of integers \(\{1, 2, \ldots, n\}\). We randomly choose a function \(f: [n] \rightarrow [n]\) out of the \(n^n\) possible functions. We also choose an integer \(a\) uniformly at random from \([n]\). Find the probability that there exist positive integers \(b, c \geq 1\) such that \(f^b(1) = a\) and \(f^c(a) = 1\). (\(f^k(x)\) denotes the result of applying \(f\) to \(x\) \(k\) times).

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