
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 the vertices of a regular tetrahedron each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following biased rule: at each vertex it chooses one of the three edges meeting at that vertex to crawl along. The bug has a preference: it selects the edge leading back to the vertex from which it just came with a probability of $\frac{1}{2}$ if possible, and splits the remaining probability equally among the other available edges. Compute the probability $p = \frac{n}{2187}$ that the bug is at vertex $A$ after crawling exactly 8 meters.

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