
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 gymnastics competition, each gymnast shook hands once with every gymnast from every team, except with themselves. After the event, two coaches each came down and shook hands with every gymnast from their respective teams. There were a total of 325 handshakes. What is the fewest number of handshakes the coaches could have participated in combined?

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