
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
After another gymnastics meet, there were two teams, Team A and Team B. Each gymnast shook hands once with every other gymnast present (except herself). Afterwards, each team's coach came down and only shook hands with each gymnast from their respective team. There were a total of 465 handshakes. What is the fewest number of handshakes the coach of the smaller team could have participated in?

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