Eccentricity from a Minor-Axis Chord

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $(a>b>0)$ shown has left and right foci $F_1$ and $F_2$. Points $A$ and $C$ on the ellipse are symmetric about the $x$-axis, $B$ is an endpoint of the minor axis, and the segment $AB$ passes through the right focus $F_2$.

Given that $AB\perp CF_1$, find the eccentricity of the ellipse.