
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, and each edge measures 1 meter. A bug starts at vertex $A$ and follows new rules: at each vertex, it chooses the next vertex based on new probabilities - it has a $\frac{1}{2}$ probability of choosing one of the two vertices that are not the immediate opposite, and a $\frac{1}{6}$ probability of choosing the opposite vertex. Calculate the probability that the bug is back at vertex $A$ after crawling exactly 6 meters.

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