
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 a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along an edge of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.

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