
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
Three mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 9:30 a.m., and each stays for exactly $n$ minutes. The probability that any one arrives while another is in the cafeteria is $50 \%,$ and $n = d - e\sqrt {f},$ where $d, e,$ and $f$ are positive integers, and $f$ is not divisible by the square of any prime. Find $d + e + f.$

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