
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
Let $A$, $B$, $C$, and $D$ be vertices of a regular tetrahedron with each edge measuring 1 meter. A bug, starting at vertex $A$, follows these rules: at each vertex, it randomly chooses one of the three connecting edges, each with an equal likelihood, and then moves to the opposite vertex. Let $p = \frac{n}{2187}$ be the probability that the bug is back at vertex $A$ after traveling exactly 8 meters. Find the value of $n$.

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