Let A B C be a triangle. The angle bisector of \angle B intersects A C at point P, while the angle bisector of \angle C intersects A B at a point Q. Suppose the area of \triangle A B P is 27, the area of \triangle A C Q is 32, and the area of \triangle A B C is 72. The length of \overline{B C} can be written in the form m \sqrt{n} where m and n are positive integers with n as small as possible. What is m+n?