
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 eleven people, each person shakes hands with exactly two of the other people from the group. Let \( N \) be the number of ways this handshaking can occur. Different handshaking arrangements are considered different if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when \( N \) is divided by 1000.

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