
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
Let  $ S$  be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer  $ k$  for which there exists a 60-element subset of set  $ S$  with the following condition satisfied for any two elements   $ A,B$  of the subset there exists a point  $ C$  contained in   $ S$  such that the area of triangle  $ ABC$  is equal to k .

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