
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, with each edge measuring 1 meter. A bug starts from vertex $A$ and follows a new rule: at each vertex, it chooses one of the three edges to proceed, with the edge leading back to the previous vertex being half as likely to be chosen as each of the other two edges. Calculate the probability that the bug returns to vertex $A$ after crawling exactly 6 meters. Let this probability be expressed as $q = \frac{m}{729}$.

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