
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
At a sports tournament, each athlete shook hands once with every athlete from all teams except themselves. Afterwards, the event ended with two coaches, each shaking hands only with the athletes from their respective teams. There were a total of 465 handshakes in all. What is the fewest number of handshakes the coaches could have participated in?

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