
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 a regional gymnastics meet, each gymnast shook hands once with every other gymnast. After the meet, two coaches, each from different teams, shook hands with every gymnast of their respective teams but did not shake hands with each other or with gymnasts from the other team. There were a total of 465 handshakes. What is the fewest number of gymnasts that could have been at the meet if the number of gymnasts in each team is the same?

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