An Intersection Locked on a Line

Let $A$ and $B$ be the left and right vertices of the ellipse $E:\dfrac{x^2}4+y^2=1$, and let $M(m,0)$ ($m>0$) be a point whose minimum distance to points of the ellipse equals $1$.

1. Find the coordinates of $M$. 2. A line $l$ through $M$ meets $E$ at two points $C$ and $D$ (distinct from $A$ and $B$); the lines $AC$ and $BD$ meet at a point $G$. (a) Prove that $G$ lies on a fixed line. (b) Does there exist a position of $G$ with $CG\perp DG$? If so, find the slope of $l$; if not, explain why.