Four circles, no two of which are congruent, have centers at $A,B,C,D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{1}{2}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{1}{2}$ times the radius of circle $D$. Furthermore, $AB=CD=15$ and $PQ=16$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$?