
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
Assume the four vertices of a regular tetrahedron are \(A, B, C, D\), and the length of each edge is 1 meter. A bug starts at point \(A\) and moves according to the following rule: At each vertex, it chooses one of the three edges connected to that vertex with equal probability and crawls to the other end of that edge. After crawling for 7 meters, what is the probability \(p\) that the bug is exactly at vertex \(A\)? Given the probability can be expressed as \( p = \frac{n}{729} \), find the value of \( n \).

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