HMMT 2013 G6

Let $A B C D$ be a quadrilateral such that $\angle A B C=\angle C D A=90^{\circ}$, and $B C=7$. Let $E$ and $F$ be on $B D$ such that $A E$ and $C F$ are perpendicular to $B D$. Suppose that $B E=3$. Determine the product of the smallest and largest possible lengths of $D F$.