In rectangle $ABCD,$ $AD=10$ and $CD=5$. Point $E$ on $CD$ and point $F$ on $DE$ are such that $CE=EF=FD$. Segments $\overline{BE}$ and $\overline{BF}$ intersect $\overline{AC}$ at $P$ and $Q$, respectively. The ratio $AQ:QP:PC$ can be written as $r:s:t$ where the greatest common factor of $r,s,$ and $t$ is $1.$ What is $r+s+t$?