
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 can either move to an adjacent vertex or diagonally to the opposite vertex. The probability of moving to an adjacent vertex is $\frac{1}{3}$ and diagonally is $\frac{1}{3}$. If another vertex is selected, it does not return to the vertex it just visited. Given that the probability the bug returns to its starting vertex on its tenth move is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, determine $p + q$.

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