
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
Consider a mapping \( f \) from an \( n \)-element set \( X \) to itself, where \( n \geq 2 \). If \( a \) is a fixed element in \( X \) and for every \( x \in X \), it holds that \( f(f(x)) = a \). Find the number of such mappings \( f \).

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