In triangle A B C, A C=13, B C=14, and A B=15. Points M and D lie on \overline{A C} with A M=M C and \angle A B D=\angle D B C. Points N and E lie on \overline{A B} with A N=N B and \angle A C E=\angle E C B. Let P be the other point of intersection of the circumcircles of \triangle A M N and \triangle A D E. Ray A P meets \overline{B C} at Q. The ratio \frac{B Q}{C Q} can be written in the form \frac{m}{n}, where m and n are relatively prime positive integers. Find m-n.