
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
In a regular tetrahedron $A B C D$ with an edge length of 1 meter, a bug starts crawling from vertex $A$. At each vertex, the bug randomly chooses one of the three edges and crawls to the end of that edge with equal probability. Let $P_{n}$ be the probability that the bug returns to vertex $A$ after crawling $n$ meters. Find $P_{4}$.

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