
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 corporate event, 10 people attend. If everyone is supposed to shake hands with everyone else but three handshakes already took place unknowingly as secret partners met—their identities remain undisclosed—how many more handshakes need to occur for everyone to have shaken hands with each other?

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