
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
The integer lattice in the plane is colored with 3 colors. Find the least positive real  $S$  with the property: for any such coloring it is possible to find a monochromatic lattice points  $A,B,C$  with  $S_{\triangle ABC}=S$ .

*Proposed by Nikolay Beluhov*

EDIT: It was the problem 3 (not 2), corrected the source title.

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