
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 gather for a convention. Similar to previous conventions, the imps have internal rivalries and will not shake hands among themselves, but they will shake hands with any gremlin. At this convention, however, one-third of the gremlins (exactly 10) are feeling particularly mischievous and decide not to shake hands with any imps. Each gremlin shakes hands with all other gremlins. Each pair of creatures shakes hands only once. How many total handshakes occur at this convention?

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