An Angle Bisector and Two Moving Points on a Ray

As shown in the figure, in the plane quadrilateral $ABCD$ we have $AB\perp BC$, $AD\perp DC$, $AB=AD=1$, and $\angle BAD=\dfrac{2\pi}3$. Two moving points $E$ and $F$ lie on ray $BC$, where $E$ lies on segment $BC$ (with $E$ distinct from $B$ and $C$), and they satisfy the condition that $DC$ bisects $\angle EDF$. Find the value of $\tan\angle EDF$ at the moment when $4BE+BF$ attains its minimum.