For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2 y}$. The sum of the solutions to the equation \[ 4096 f(f(x, x), x) = x^{13} \] can be written in simplest form as $\frac{m}{n}$. Compute $m + n$.

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