
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
A moving point $P$ starts from vertex $A$ of the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, moves along the edges to vertex $C_{1}$ and then back to $A$. If the path exactly passes through $6$ different edges, such a path is called the "optimal path". The number of "optimal paths" is ______. (Answer with a number).

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