
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's choice, $a$, is uniformly random over [0,1], and Bob's choice, $b$, is uniformly random over [0.25, 0.75], determine the probability that Carol's choice will maximize her chances of being exactly in between Alice's and Bob's numbers.

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