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 $m+n+p$.
1. **Define the circles and their properties:**
   - Let the radius of circle \( C_1 \) be \( r_1 = 4 \).
   - Let the radius of circle \( C_2 \) be \( r_2 = 10 \).
   - Let the radius of circle \( C_3 \) be \( R \).

2. **Determine the radius of \( C_3 \):**
   - Since \( C_1 \) and \( C_2 \) are internally tangent to \( C_3 \) and externally tangent to each other, the distance between the centers of \( C_1 \) and \( C_2 \) is \( r_1 + r_2 = 4 + 10 = 14 \).
   - The distance from the center of \( C_3 \) to the center of \( C_1 \) is \( R - r_1 \).
   - The distance from the center of \( C_3 \) to the center of \( C_2 \) is \( R - r_2 \).
   - Since the centers are collinear, the sum of these distances is equal to the distance between the centers of \( C_1 \) and \( C_2 \):
     \[
     (R - 4) + (R - 10) = 14
     \]
     \[
     2R - 14 = 14
     \]
     \[
     2R = 28
     \]
     \[
     R = 14
     \]

3. **Set up the coordinate system:**
   - Place the center of \( C_3 \) at the origin \((0,0)\).
   - The center of \( C_1 \) is at \((10,0)\) because it is 10 units to the right of the origin.
   - The center of \( C_2 \) is at \((-4,0)\) because it is 4 units to the left of the origin.

4. **Find the equation of the chord:**
   - The chord is a common external tangent to \( C_1 \) and \( C_2 \).
   - The distance between the centers of \( C_1 \) and \( C_2 \) is 14 units.
   - The length of the chord can be found using the Pythagorean theorem in the right triangle formed by the radius of \( C_3 \), the distance from the center of \( C_3 \) to the chord, and half the length of the chord.

5. **Calculate the length of the chord:**
   - The equation of \( C_3 \) is \( x^2 + y^2 = 196 \).
   - The chord is vertical and its equation is \( x = -6 \) (since it is the midpoint between the centers of \( C_1 \) and \( C_2 \)).
   - Substitute \( x = -6 \) into the equation of \( C_3 \):
     \[
     (-6)^2 + y^2 = 196
     \]
     \[
     36 + y^2 = 196
     \]
     \[
     y^2 = 160
     \]
     \[
     y = \pm 4\sqrt{10}
     \]
   - The length of the chord is the distance between \( y = 4\sqrt{10} \) and \( y = -4\sqrt{10} \):
     \[
     \text{Length of the chord} = 2 \times 4\sqrt{10} = 8\sqrt{10}
     \]

6. **Express the length in the required form:**
   - The length of the chord is \( \frac{8\sqrt{10}}{1} \).
   - Here, \( m = 8 \), \( n = 10 \), and \( p = 1 \).

7. **Sum the values:**
   - \( m + n + p = 8 + 10 + 1 = 19 \).

The final answer is \( \boxed{19} \).