2003 AIME II Problem 11

Triangle A B C is a right triangle with A C=7, B C=24, and right angle at C. Point M is the midpoint of \overline{A B}, and D is on the same side of line A B as C so that A D=B D=15. Given that the area of \triangle C D M can be expressed as \frac{m \sqrt{n}}{p}, where m, n, and p are positive integers, m and p are relatively prime, and n is not divisible by the square of any prime, find m+n+p.