
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
Jane, Bob, and Alice are playing a game in which they take turns throwing a ball at a bottle, with Jane throwing first, then Bob, then Alice, and then back to Jane, in a cyclic order. The probability that Jane or Alice knocks the bottle off on their turn is $\frac{1}{3}$, while the probability that Bob knocks it off is $\frac{2}{3}$. Calculate the probability that Jane wins the game.

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