
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 starts at vertex $A$ and at each vertex, it randomly chooses one of the three connecting edges to crawl to the next vertex. The probability of choosing any edge is equal. Determine the probability that the bug is at vertex $A$ after crawling exactly 8 meters, expressed in the form $p = \frac{n}{2187}$, and find $n$.

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