
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 a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its twelfth move is \( \frac{m}{n} \), 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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