
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 \( A = \{1, 2, \cdots, 10\} \). The mapping \( f \) from \( A \) to \( A \) satisfies the following two conditions:
1. For any \( x \in A \), \( f_{30}(x) = x \).
2. For every \( k \in \mathbf{N}_{+}, 1 \leq k \leq 29 \), there exists at least one \( a \in A \) such that \( f_{k}(a) \neq a \).

Determine the number of such mappings \( f \) (where it is given \( f_{1}(x) = f(x), f_{k+1}(x) = f(f_{k}(x)) \) for \( k = 1, 2, \cdots \)).

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