
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
Thirty gremlins and twenty imps are at the Interregional Mischief Convention. This year, five of the imps, due to a magical disagreement, refuse to shake hands with ten specific gremlins. The rest of the imps have no issues shaking hands with all gremlins. All gremlins are still friendly and shake hands with each other and with any imp willing to shake hands with them. How many handshakes were at the convention?

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