
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
A bug starts at vertex  $A$  of triangle  $ABC$ . Six times, the bug travels to a randomly chosen adjacent vertex. For example, the bug could go from  $A$ , to  $B$ , to  $C$ , back to  $B$ , and back to  $C$ . What is the probability that the bug ends up at  $A$  after its six moves?

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