
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, Carol, and Dave play a game where each chooses a real number within specified intervals. Alice chooses her number uniformly at random from all numbers between $0$ and $1$, Bob from $\frac{1}{3}$ to $\frac{2}{3}$, and Dave from $0$ to $\frac{1}{3}$. What number should Carol choose to maximize her chance of winning, where the winner's number is between the numbers chosen by the other three players?
A) $\frac{1}{4}$
B) $\frac{13}{24}$
C) $\frac{7}{12}$
D) $\frac{5}{8}$
E) $\frac{2}{3}$

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