
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, each person shakes hands with exactly three other people from the group. Find the remainder of $N$, the number of ways this handshaking can occur, when divided by 1000. Assume two handshaking arrangements are different if there is at least one pair of people who shake hands in one arrangement but not in the other.

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