Circles \(C_1\) and \(C_2\) are externally tangent, and they are both internally tangent to circle \(C_3.\) The radii of \(C_1\) and \(C_2\) are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of \(C_3\) is also a common external tangent of \(C_1\) and \(C_2.\) Given that the length of the chord is \(\frac{m\sqrt{n}}p\) 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 \(m+n+p.\)
(第二十三届AIME2 2005 第8题)