HMMT 2019 G2

In rectangle $A B C D$, points $E$ and $F$ lie on sides $A B$ and $C D$ respectively such that both $A F$ and $C E$ are perpendicular to diagonal $B D$. Given that $B F$ and $D E$ separate $A B C D$ into three polygons with equal area, and that $E F=1$, find the length of $B D$.