Circle C with radius 2 has diameter \overline{A B}. Circle D is internally tangent to circle C at A. Circle E is internally tangent to circle C, externally tangent to circle D, and tangent to \overline{A B}. The radius of circle D is three times the radius of circle E and can be written in the form \sqrt{m}-n, where m and n are positive integers. Find m+n.