
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 coordinate plane, consider the set \( S \) of all points with integer coordinates. For a positive integer \( k \), two distinct points \( A, B \in S \) will be called \( k \)-friends if there is a point \( C \in S \) such that the area of the triangle \( A B C \) is equal to \( k \). A set \( T \subset S \) will be called a \( k \)-clique if every two points in \( T \) are \( k \)-friends. Find the least positive integer \( k \) for which there exists a \( k \)-clique with more than 200 elements.

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