There are positive integers $m$ and $n$ such that $m^2 -n = 32$ and  $\sqrt[5]{m +\sqrt{n}}+  \sqrt[5]{m -\sqrt{n}}$ is a real root of the polynomial $x^5 - 10x^3 + 20x - 40$. Find $m + n$.

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