
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 the set \( S = \{ x \mid x \) is a 9-digit number in decimal representation, with the digits being \( 1, 2, 3 \} \). Define the mapping \( f: S \longrightarrow \{1, 2, 3\} \), such that for any pair of numbers \( x \) and \( y \) in \( S \) with all digits at the same positions being different, \( f(x) \neq f(y) \). Find the mapping \( f \) and the number of such mappings.

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