Circles 𝐶1 and 𝐶2 are externally tangent, and they are both internally tangent to circle 𝐶3. The radii of 𝐶1 and 𝐶2 are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of 𝐶3 is also a common external tangent of 𝐶1 and 𝐶2. 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.
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