2014 AIME II Problem 8

Circle C with radius 2 has diameter \overline{AB}. 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{AB}. 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.