
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 gymnastics event, gymnasts shook hands once with each gymnast from both their own team and the opposing team. Thereafter, each of two coaches, one for each team, shook hands only with their own team members. The total number of handshakes was 620. If Team A has 2 more gymnasts than Team B, and assuming the fewest possible handshakes among the coaches, what is the fewest number of handshakes that the coaches from Team A and Team B participated in?

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