
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
In a 3-dimensional Cartesian coordinate system, the set of points \( S = \{(x, y, z) \mid x, y, z \in \{0, 1\}\} \). A bijection \( f: S \rightarrow S \) satisfies \( |Af(A)| = 1 \) for any \( A \in S \). Determine the number of such bijections \( f \).

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