
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 12 people, each person shakes hands with exactly three of the other people from the group. Let $M$ be the number of ways this handshaking can occur, with two arrangements considered different if there is at least one person who shakes hands with a different group of three people under the other arrangement. Find the remainder when $M$ is divided by 1000.

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