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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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