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?