
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<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub> and moves along the edges to vertex C<sub>1</sub> 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 in digits).

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