
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  $n$  be a positive integer.  $S_1, S_2, \ldots, S_n$  are pairwise non-intersecting sets, and  $S_k $  has exactly  $k$  elements  $(k = 1, 2, \ldots, n)$ . Define  $S = S_1\cup S_2\cup\cdots \cup S_n$ . The function  $f: S \to S $  maps all elements in  $S_k$  to a fixed element of  $S_k$ ,  $k = 1, 2, \ldots, n$ . Find the number of functions  $g: S \to S$  satisfying  $f(g(f(x))) = f(x).$ 

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