
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 go for a swimming session at a local pool every day. They arrive independently at random times between 3 p.m. and 4 p.m. and each stays for exactly $n$ minutes. The probability that either friend arrives while the other is at the pool is $50 \%,$ and $n = p - q\sqrt {r},$ where $p, q,$ and $r$ are positive integers, and $r$ is not divisible by the square of any prime. Find $p+q+r.$

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