Lattice Points

A lattice point is defined as a point in the two-dimensional plane with integral coordinates. We define the centroid of four points (xi,yi),i=1,2,3,4(x_i,y_i), i=1,2,3,4 as point (x1+x2+x3+x44,y1+y2+y3+y44).\left(\frac{x_1+x_2+x_3+x_4}4,\frac{y_1+y_2+y_3+y_4}4\right).

What is the largest number of distinct lattice points in the plane such that the centroid of any four of them is not a lattice point?

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