
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
Compute the number of functions  $f\colon\{1, \dots, 15\} \to \{1, \dots, 15\}$  such that, for all  $x \in \{1, \dots, 15\}$ , \[
    \frac{f(f(x)) - 2f(x) + x}{15}
\]is an integer. 

*Proposed by Ankan Bhattacharya*

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