Let \(m\) be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\]

There are positive integers \(a,b,c\) such that \(m = a + \sqrt{b+\sqrt{c}}\). Find \(a+b+c\).

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