Consider isosceles right triangle $ABC$ with $BC$ as the hypotenuse and $AB=AC=6$. $E$ is on $BC$ and $F$ is on $AB$ such that $AE+EF+FC$ is minimized. Compute $EF$.

My thought process:

I reflected triangle $ABC$ across $BC$ to get a square $ABA'C$. Then, I messed around with the placement of $F$ on $AB$, to see what results I could produce, with $E$ always being the midpoint. May someone help me on this?