
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
In a group of twelve people where each person shakes hands with exactly three others, find the number of distinct handshaking arrangements, modulo 1000. The arrangement is considered different if at least one pair of people who shake hands in one arrangement do not do so in another.

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