2011 AIME II Problem 8

Let z_{1}, z_{2}, z_{3}, \ldots, z_{12} be the 12 zeros of the polynomial z^{12}-2^{36}. For each j,
let w_{j} be one of z_{j} or i z_{j}. Then the maximum possible value of the real part of
\sum_{j=1}^{12} w_{j} can be written as m+\sqrt{n}, where m and n are positive integers. Find m+n.