Let $z_1,z_2,z_3,\dots,z_{12}$ be the $12$ zeroes 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 $\sum\limits_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.

Let's think step by step and output the final answer within \boxed{}.
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