2011 AIME II Problems/Problem 8

Problem:

Let $z_{1}, z_{2}, ... , z_{12}$ 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 (somebody who knows how please create in an equation) SUM j=1 to 12 ( $w_{j}$ ) can be written as m+root(n), where m and n are positive integers. Find m+n.