
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 Carol once again decide to play a number choosing game where each picks a number between $0$ and $1$. The winner is the one whose number is between the numbers chosen by the other two. Alice still picks her number uniformly from the interval $[0,1]$. This time, Bob changes his strategy and decides to pick his number uniformly at random between $\frac{1}{3}$ and $\frac{3}{4}$. What number should Carol choose to maximize her chance of winning?

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