
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 the set \( A = \{1, 2, \cdots, 6\} \), the mapping \( f: A \rightarrow A \) is a bijection,
$$
\begin{array}{l}
f^{(2)}(x) = f(f(x)), \\
f^{(n+1)}(x) = f\left(f^{(n)}(x)\right) \quad (n \geqslant 1).
\end{array}
$$

The number of mappings \( f \) that satisfy \( f^{(12)}(x) = x \) for any \( x \in A \) is .

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