
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 basketball tournament, each player shook hands once with every player from other teams only and not with their own team members. After the games, two coaches (one from each of two teams) shook hands with every player from their respective teams. There were a total of 495 handshakes. What is the fewest number of handshakes that the coaches could have participated in?

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