Avoid lattice midpoints

Geometry Level 2

You select a set of \(N\) specific points in the 2D plane with integer coordinates such that, for every pair of the \(N\) points, the midpoint of the segment connecting them does not have integer coordinates.

What is the greatest possible value of \(N?\)

Below is an invalid selection of 3 points (in red), since the midpoint of \(BC\) has integer coordinates.

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