
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 twelve pixies are at the Forest Gathering. Due to a recent dispute, half of the gremlins (15 gremlins) refuse to shake hands with each other but will shake hands with the pixies and the other half of the gremlins. The rest of the gremlins are friendly and shake hands with all gremlins and all pixies. The pixies, neutral in the disputes, shake hands with all gremlins. Each pair of creatures shakes hands at most once. How many handshakes occur at this gathering?

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