
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 historical commemoration event, $15$ married couples gather where each man shakes hands only with other men and women (excluding his own wife), and each woman also shakes hands with all other women (excluding handshakes with men). Calculate the total number of handshakes among these $30$ people.

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