
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
Alice and Bob modify their game with a focus on improving their passing skills. Now, if Alice has the ball, there is a 2/3 chance she will pass it to Bob and a 1/3 chance she will keep it. If Bob has the ball, there is a 1/2 chance he will pass it back to Alice, otherwise, he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after two turns?

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