
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
Two mathematicians take a morning break each day, arriving independently at the cafeteria at random times between 9 a.m. and 11 a.m. They stay for exactly \( m \) minutes. The probability that either arrives while the other is in the cafeteria is \( 50\% \). Find \( m \) if it is expressed as \( m = a - b\sqrt{c} \), where \( a, b, c \) are positive integers, and \( c \) is not divisible by the square of any prime. Determine \( a + b + c \).

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