$x^4 - 2x^2 - 8x - 3 = 0$

Let $a,b,c$ and $d$ denote the roots of the equation above. And suppose we denote $2\sqrt n + 2\sqrt m = |a| + |b| + |c| + |d|$ for positive integers $m$ and $n$, find the value of $m^2 + n^2$.

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