
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 of a cube and moves along the edges of the cube under the rule that at each vertex the bug will choose one of the three emanating edges with equal probability. All moves are independent. What is the probability that after nine moves the bug will have visited every vertex at least once?

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