
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
Consider a regular octahedron (a polyhedron consisting of 8 congruent equilateral triangular faces) suspended in space with one face on top and one directly below it, which are horizontal. The top face is surrounded by a ring of three slanted faces, and similarly, the bottom face is also surrounded by a ring of three slanted faces, each adjacent to the bottom face. Determine the number of ways to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring.

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