Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt{6}$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt{3}$. Find the length of $|AC|$.

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