A lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the plane, find the smallest number of line segments $\overline{AB}$ for which $A$ and $B$ are distinct lattice points in this set, and the midpoint of $\overline{AB}$ is also a lattice point (not necessarily in the set).

Let's think step by step and output the final answer within \boxed{}.
user