Incircle of a Triangle Inscribed in an Ellipse

The circle $G:(x-2)^2+y^2=r^2$ is the incircle of triangle $ABC$, which is inscribed in the ellipse $\dfrac{x^2}{16}+y^2=1$, where $A$ is the left vertex of the ellipse.

(1) Find the radius $r$ of circle $G$.

(2) Through $M(0,1)$ draw the two tangent lines to the circle, meeting the ellipse again at points $E$ and $F$. Prove that line $EF$ is tangent to circle $G$.