Let \triangle A B C be a triangle with B C=7, C A=6, and, A B=5. Let I be the incenter of \triangle A B C. Let the incircle of \triangle A B C touch sides B C, C A, and A B at points D, E, and F. Let the circumcircle of \triangle A E F meet the circumcircle of \triangle A B C for a second time at point X \neq A. Let P denote the intersection of X I and E F. If the product X P \cdot I P can be written as \frac{m}{n} for relatively prime positive integers m, n, find m+n.