
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 \( A = \{1,2,3,4,5,6,7,8,9,10,11\} \). \( f \) is a mapping from \( A \) to \( A \) that satisfies:
1. For any \( x \in A \), \( f_{15}(x) = x \);
2. For any positive integer \( k \) ( \( 1 \leq k \leq 14 \) ), there is at least one \( a \in A \) such that \( f_k(a) \neq a \).

Find the number of mappings \( f \) that satisfy the above conditions. Here \( f_1(x) = f(x) \) and \( f_{k+1}(x) = f(f_k(x)) \) for \( x \in A \), \( k = 1, 2, 3, \ldots \).

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