2020 AMC 10B Question 20

Let $B$ be a right rectangular prism (box) with edge lengths $1,$ $3,$ and $4,$ together with its interior. For real $r\geq0,$ 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 $$
ar^{3} + br^{2} + cr +d,
$$ where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\dfrac{bc}{ad}?$