Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C = \frac{1}{3} BD$. If the area of $ABCD$ is $27\sqrt{2}$, find $BD$.

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