
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 play an extended game with a soccer ball. On each turn, if Alice has the ball:
- There is a 1/3 chance she will toss it to Bob,
- A 1/3 chance she will pass it back to a team member who gives it back to her,
- A 1/3 chance she will keep the ball. 

If Bob has the ball, there is:
- A 3/5 chance he will toss it back to Alice, 
- A 2/5 chance he will keep it. 

Alice starts with the ball. What is the probability that Alice has the ball after three turns?

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