Rectangle A B C D and a semicircle with diameter \overline{A B} are coplanar and have nonoverlapping interiors. Let \mathcal{R} denote the region enclosed by the semicircle and the rectangle. Line \ell meets the semicircle, segment \overline{A B}, and segment \overline{C D} at distinct points N, U, and T, respectively. Line \ell divides region \mathcal{R} into two regions with areas in the ratio 1: 2. Suppose that A U=84, A N=126, and U B=168. Then D A can be represented as m \sqrt{n}, where m and n are positive integers and n is not divisible by the square of any prime. Find m+n.