Let $B$ be a cube with edge length $2$, together with its interior. For real $r\ge 0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $A\pi r^3+B\pi r^2+Cr+D$, where $A,B,C,$ and $D$ are positive real numbers. What is $A+B+C+D$?