Eccentricity from a Perpendicular Focal 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 $F_2$. A line through $F_2$ meets the ellipse at points $P$ and $Q$, and $PQ\perp PF_1$.

Given that $|PQ|=\lambda|PF_1|$ where $\dfrac{3}{4}\leqslant\lambda<\dfrac{4}{3}$, determine the range of the eccentricity $e$ of the ellipse.