
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 two vertices where it is not currently located, and moves directly to that vertex. Determine the probability that the bug returns to its starting vertex on its eighth move. Express this probability as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m + n.$

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