Let C1 be a circle with center A and radius 1. Let m be a line tangent to circle C1 at point B. Let C2 be a circle with center B intersecting line m at points C and D and circle C1 at points E and F (points labeled such that C and E are on the same side of line AB). Let line CE intersect line AB at point G. As the radius of circle C2 shrinks to zero, does the length of BG approach a limit? If yes, then what is its value?