
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 students, A and B, each decide to study in the library. They arrive independently at random times between 9 a.m. and 11 a.m., and each stay for exactly $n$ minutes. The probability that one arrives while the other is still in the library is $50\%$. Express $n$ in the form $n = x - y\sqrt{z}$, where $x, y,$ and $z$ are positive integers, and $z$ is not divisible by the square of any prime. Find $x + y + z$.

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