A triangle $\triangle ABC$ has a right angle at $C$. The lengths of the legs $AC$ and $BC$ are equal, and the length of the hypotenuse $AB$ is $10$ units. A point $D$ lies on $AC$ such that $BD$ is perpendicular to $AC$. Find the area of $\triangle ABD$. Let's think step by step and output the final answer within \boxed{}.
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