
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
A lattice point is a point \((x, y)\) with both \(x\) and \(y\) integers. Find, with proof, the smallest \(n\) such that every set of \(n\) lattice points contains three points that are the vertices of a triangle with integer area. (The triangle may be degenerate; in other words, the three points may lie on a straight line and hence form a triangle with area zero.)

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