A Triple-Slope Ray

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{2\sqrt{2}}{3}$, bottom vertex $A$, right vertex $B$, and $|AB| = \sqrt{10}$.

1. Find the standard equation of the ellipse $C$. 2. A moving point $P$ lies off the $y$-axis, and a point $R$ on the ray $AP$ satisfies $|AP|\cdot|AR| = 3$.

1. With $P(m, n)$, find the coordinates of $R$ in terms of $m$ and $n$.
2. Let $O$ be the origin and let $Q$ be a moving point on $C$. If the slope of line $OR$ is $3$ times the slope of line $OP$, find the maximum value of $|PQ|$.