Problem Analysis #8

Return a single integer only. The problem has been revised so that no final answer is an irrational number, a parameterized expression, a range, a proof, a drawing, a graph, a locus equation, or a region description.

Let $A(1,0,0)$, $B(0,1,0)$, $C(0,0,1)$, and let $D$ be the midpoint of $AC$. Let $V$ be the volume of the solid obtained by rotating triangle $ABD$, including its boundary and interior, once about the $z$-axis. If $V=p\pi/q$ in lowest terms, find $p+q$.