HMMT 2012 G2

Let $A B C$ be a triangle with $\angle A=90^{\circ}, A B=1$, and $A C=2$. Let $\ell$ be a line through $A$ perpendicular to $B C$, and let the perpendicular bisectors of $A B$ and $A C$ meet $\ell$ at $E$ and $F$, respectively. Find the length of segment $E F$.