2005 AIME II Problem 8

Circles \mathcal{C}_{1} and \mathcal{C}_{2} are externally tangent, and they are both internally tangent to circle \mathcal{C}_{3}. The radii of \mathcal{C}_{1} and \mathcal{C}_{2} are 4 and 10, respectively, and the centers of the three circles are collinear. A chord of \mathcal{C}_{3} is also a common external tangent of \mathcal{C}_{1} and \mathcal{C}_{2}. Given that the length of the chord is 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.