
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, Bob, and Clara play a game with a baseball. On each turn:
- If Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball.
- If Bob has the ball, there is a 1/3 chance he will toss it to Alice, a 1/3 chance he will toss it to Clara, and a 1/3 chance that he will keep it.
- If Clara receives the ball, she passes it to Alice with a probability of 1/2 or keeps it also with a probability of 1/2.
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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