
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
Andrew the ant starts at vertex  $A$  of square  $ABCD$ . Each time he moves, he chooses the clockwise vertex with probability  $\frac{2}{3}$  and the counter-clockwise vertex with probability  $\frac{1}{3}$ . What is the probability that he ends up on vertex  $A$  after  $6$  moves?

*2015 CCA Math Bonanza Lightning Round #4.3*

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