
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
Given that Alice chooses her number uniformly at random from all numbers between $0$ and $1$, Bob chooses his number uniformly at random from all numbers between $\frac{1}{3}$ and $\frac{2}{3}$, and Carol chooses her number uniformly at random from an unknown interval, determine the interval for Carol's number that will maximize her probability of being the winner.

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