
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 coffee break each day.  They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly  $m$  mintues.  The probability that either one arrives while the other is in the cafeteria is  $40 \%,$  and  $m=a-b\sqrt{c},$  where  $a, b,$  and  $c$  are positive integers, and  $c$  is not divisible by the square of any prime.  Find  $a+b+c.$ 

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