Let $ABCD$ be a unit square. Let $Q_1$ be a point on $\overline{CD}$ such that $D{Q}_1=\frac{1}{3}$. For $i=1,2,\dots$, let $P_i$ be the intersection of $\overline{A{Q}_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is