
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 the corners of a square courtyard stand four houses, each inhabited by hooligans who are friends with each other. Starting from January 1, 2017, every day two hooligans from neighboring houses would permanently fall out with each other, and by January 1, 2018, it was found that there were no more pairs of hooligans left to fall out. How many hooligans could there have been in total? List all possible variants and explain why there are no others.

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