
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  $ ABC$  is equal to  $ k$ . A set  $ T\subset S$  will be called  $ k$ -*clique* if every two points in  $ T$  are  $ k$ -friends. Find the least positive integer  $ k$  for which there exits a  $ k$ -clique with more than 200 elements.

*Proposed by Jorge Tipe, Peru*

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