Let $ABCD$ be a quadrilateral with sides $|\overline{AB}|=2$, $|\overline{BC}|=|\overline{CD}|=4$, and $|\overline{DA}|=5$. The opposite angles, $\angle A$ and $\angle C$, are equal. What is the length of diagonal $BD$? The original answer is in $k\sqrt{m}$ format, please give the value of k + m.

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