A Reflected Focal Chord

A moving line parallel to the $x$-axis meets the hyperbola $E \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) at points $A$ and $B$, with $A$ to the left of $B$. Let $F$ be the left focus of $E$. Extend $BF$ to a point $C$ with $|AF| = |FC|$, and let line $AC$ meet the $x$-axis at $D$. If $|FC| = 3|FD|$, find the eccentricity of the hyperbola.