
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]


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In a group of twelve people, each person shakes hands with three different individuals. Calculate the number of distinct handshaking arrangements, assuming that an arrangement is considered different if there is at least one pair of individuals who shake hands in one arrangement but not in another arrangement. Then find the remainder when this number is divided by $1000$.

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