A Dot-Product Condition on an Ellipse

In the coordinate plane, consider the ellipse $\Gamma \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Let $A$ be an endpoint of the major axis, $B$ an endpoint of the minor axis, and $F$ a focus of $\Gamma$. Suppose there exist points $P, Q$ on $\Gamma$, symmetric about the origin $O$, such that

1. Prove that the focus $F$ lies on the extension of segment $AO$ beyond $O$. 2. Find the range of possible eccentricities of $\Gamma$.