
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 friends plan to meet at a park for a walk. They both decide to come at random, independent times between 3 p.m. and 5 p.m. They will stay for different durations, $m_1$ and $m_2$, respectively. The probability that either one arrives while the other is already there is $50\%$. If $m_1 + m_2 = a - b\sqrt{c}$, where $a$, $b$, and $c$ are integers, $c$ is not divisible by the square of any prime, and $m_1 = m_2$, find the value of $a + b + c$.

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