2019 AIME I Problem 12

Given f(z) = z^2-19z, there are complex numbers z with the property that z, f(z), and f(f(z)) are the vertices of a right triangle in the complex plane with a right angle at f(z). There are positive integers m and n such that one such value of z is m+\sqrt{n}+11i. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.