
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 continue playing their game. Now, if Alice has the ball, she has a 1/3 chance of keeping it and a 2/3 chance of passing it to Bob. If Bob has the ball, there is a 1/2 chance he will pass it back to Alice. Alice still starts with the ball. What is the probability that Alice has the ball again after three turns?

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