
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. Every move, it randomly selects one of the three vertices it is not currently located at, and crawls along an edge to that vertex. Calculate the probability the bug returns to its starting vertex on its twelfth move, expressed as a reduced fraction $m/n$. Find $m + n$.

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