
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 \( X = \{1, 2, \ldots, 10\} \). The mapping \( f: X \rightarrow X \) satisfies:
1. \( f \circ f = I_X \), where \( f \circ f \) is the composition of mappings, and \( I_X \) is the identity mapping on \( X \);
2. \( |f(i) - i| \leqslant 2 \) for any \( i \in X \).

Determine the number of such mappings \( f \).

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