
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 bug starts at one vertex (vertex A) of an octahedron and moves along the edges according to a rule where at each vertex the bug will choose to move along one of the four edges emanating from that vertex, with each edge having equal probability of being chosen. All moves are independent. What is the probability that after eight moves, the bug will end up at the opposite vertex of the starting vertex (vertex B)?

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