
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
There are twenty people in a room, with \( a \) men and \( b \) women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is 106. Determine the value of \( a \times b \).

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