An Equilateral on a Directrix

Consider the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Let $PQ$ be a focal chord through the left focus $F$, not perpendicular to the $x$-axis. Suppose there is a point $R$ on the left directrix such that $\triangle PQR$ is equilateral. Find the range of possible values of the eccentricity $e$, and express the slope of the line $PQ$ in terms of $e$.