In rectangle $ABCD,$ $AB=8$ and $BC=4$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE:EF:FC=1:2:1$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s,$ and $t$ is $1.$ What is $r+s+t$?