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Circles \( \mathrm{C}_{1} \) and \( \mathrm{C}_{2} \) are externally tangent and they are both internally tangent to the circle \( \mathrm{C}_{3} \). The radii of \( \mathrm{C}_{1} \) and \( \mathrm{C}_{2} \) are 4 and 10, respectively and the centres of the three circles are collinear. \( \mathrm{A} \) chord of \( \mathrm{C}_{3} \)
\( \mathrm{P} \)
W 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 \( (m+n+p) \).
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