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)