In a right triangle $\triangle ABC$ with $\angle B$ as the right angle, the lengths of $AB$ and $BC$ are 9 units and 12 units, respectively. A parallelogram is constructed using $\triangle ABC$ as one of its halves, with the other half being an identical right triangle $\triangle A'B'C'$ reflected over $BC$. What is the area of the parallelogram in square units? Let's think step by step and output the final answer within \boxed{}.
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