An Area Range from a Parallel Chord

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) is given. A line $l$ passes through the origin, and a line parallel to $l$ meets the ellipse $C$ at two points $A$ and $B$, where $A(2,\sqrt 2)$. When $l\perp x$-axis, the line $AB$ passes through a focus of the ellipse.

1. Find the equation of the ellipse $C$. 2. Let points $D,E,G,H$ satisfy $3\overrightarrow{OE}=\overrightarrow{OD}=-\overrightarrow{OB}$; the line $AE$ meets $l$ at a point $G$, and the line $DG$ meets $AB$ at a point $H$. Find the range of the area of $\triangle AGH$.