Geometry Level 5

Suppose $$ABCD$$ is a cyclic quadrilateral with all integral sidelengths and $$AD$$ being the circumdiameter. Sides $$\overline{AB}$$ and $$\overline{BC}$$ both having length $$a$$ and side $$CD$$ having length $$b$$ such that $$a \ne b.$$

Determine the sum of the 4 smallest possible perimeters of quadrilateral $$ABCD.$$

Inspired by this beautiful problem

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