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 $a$, $b$, and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$. Find $a+b+c$.
The first step is to notice that the 4 on the right hand side can simplify the terms on the left hand side. If we distribute 1 to $\frac{3}{x-3}$, then the fraction becomes of the form $\frac{x}{x - 3}$. A similar cancellation happens with the other four terms. If we assume $x = 0$ is not the highest solution (which is true given the answer format) we can cancel the common factor of $x$ from both sides of the equation.
$\frac{1}{x - 3} + \frac{1}{x - 5} + \frac{1}{x - 17} + \frac{1}{x - 19} = x - 11$
Then, if we make the substitution $y = x - 11$, we can further simplify.
$\frac{1}{y + 8} + \frac{1}{y + 6} + \frac{1}{y - 6} + \frac{1}{y - 8} =y$
If we group and combine the terms of the form  $y - n$ and   $y + n$, we get this equation:
$\frac{2y}{y^2 - 64} + \frac{2y}{y^2 - 36} = y$
Then, we can cancel out a $y$ from both sides, knowing that $x = 11$ is not a possible solution given the answer format.
After we do that, we can make the final substitution $z = y^2$.
$\frac{2}{z - 64} + \frac{2}{z - 36} = 1$
$2z - 128 + 2z - 72 = (z - 64)(z - 36)$
$4z -  200 = z^2 - 100z + 64(36)$
$z^2 - 104z + 2504 = 0$
Using the quadratic formula, we get that the largest solution for $z$ is $z = 52 + 10\sqrt{2}$. Then, repeatedly substituting backwards, we find that the largest value of $x$ is $11 + \sqrt{52 + \sqrt{200}}$. The answer is thus $11 + 52 + 200 = \boxed{263}$