In rectangle A B C D, A B=12 and B C=10. Points E and F lie inside rectangle A B C D so that B E=9, D F=8, \overline{B E}\|\overline{D F}, \overline{E F}\| \overline{A B}, and line B E intersects segment \overline{A D}. The length E F can be expressed in the form m \sqrt{n}-p, where m, n, and p are positive integers and n is not divisible by the square of any prime. Find m+n+p.