
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
Given a regular tetrahedron with vertices \( A, B, C, D \) and edge lengths of 1 meter, a bug starts at vertex \( A \) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to that vertex with equal probability and moves to the end of that edge. Calculate the probability \( p = \frac{n}{729} \) that the bug is at vertex \( A \) after traveling 7 meters. Find the value of \( n \).

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