Consider a set $S$ of $16$ lattice points. The $16$ points of $S$ are divided into $8$ pairs in such a way that for every point $A$ and any of the $7$ pairs of points $(B,C)$ where $A$ is not included, $A$ is at a distance of at most $\sqrt{5}$ from either $B$ or $C$ Prove that any two points in the set $S$ are at a distance of at most $3\sqrt5$.