
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 four vertices \(A, B, C, D\) and each edge of length 1 meter, a bug starts at point \(A\) 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 crawls to the other end of the chosen edge. What is the probability that the bug will be exactly at vertex \(A\) after crawling a total of 7 meters?

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