
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
 
Given a positive integer \( n \geq 3 \), and a function \( f: \{1, 2, \cdots, n\} \rightarrow \{1, 2, \cdots, n\} \) satisfying \( f(f(k)) = (f(k))^3 - 6(f(k))^2 + 12 f(k) - 6 \) for all \( k \in \{1, 2, \cdots, n\} \), find the number of such functions.

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