
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 the edges of the square to that vertex. Determine the probability that the bug returns to its starting vertex on its twelfth move. Express the probability as $m/n$ where $m$ and $n$ are relatively prime integers, and find $m+n$.

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