
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  $S = \{1, 2, \dots, 9\}.$  Compute the number of functions  $f : S \rightarrow S$  such that, for all  $s \in S, f(f(f(s))) =s$  and  $f(s) - s$  is not divisible by  $3$ .

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