
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  $\left\{ 1, 2, \ldots, n \right\}$ . We randomly choose a function  $f:[n] \to [n]$ , 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}
user