
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 \( a_{1}, a_{2}, a_{3}, a_{4} \) be any permutation of \{1, 2, 3, 4\\}. Let \( f \) be a mapping from \{1, 2, 3, 4\} to \{1, 2, 3, 4\} such that \( f(i) \neq i \) for all \( i \). Consider the table \(\left[ \begin{array}{cccc} a_{1} & a_{2} & a_{3} & a_{4} \\ f(a_{1}) & f(a_{2}) & f(a_{3}) & f(a_{4}) \end{array} \right] \). Two tables \( M \) and \( N \) are said to be different if they differ in at least one corresponding entry. Determine the number of different tables satisfying these conditions.

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