In \triangle A B C, A B=10, \angle A=30^{\circ}, and \angle C=45^{\circ}. Let H, D, and M be points on line \overline{B C} such that \overline{A H} \perp \overline{B C}, \angle B A D=\angle C A D, and B M=C M. Point N is the midpoint of segment \overline{H M}, and point P is on ray A D such that \overline{P N} \perp \overline{B C}. Then A P^{2}=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.