Question 13
Rectangle $ABCD$ and a semicircle with diameter $\overline{AB}$ 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 $\overline{AB},$ and segment $\overline{CD}$ 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 $AU = 84,$ $AN = 126,$ and $UB = 168.$ Then $DA$ 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 $m + n.$