Circles $C_1$ and $C_2$ are externally tangent and they both are internally tangent to the circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$ respectively, and the centres of the three circles are collinear. A chord of $C_3$ is also a common internal 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 the value of $\left(m+n+p\right)$.