An Eccentricity from a 120° Angle

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has right focus $F$. Let $A$ be a point on the right branch of $C$, and let $D$, $E$ be the reflections of $A$ across the origin and across the $x$-axis respectively. Given $EF\parallel AD$ and $\angle AFE=120^\circ$, find the eccentricity of $C$.