2024 Fermat Problem 14

In diagram, point C is on side BD of quadrilateral ABDE. Also, AB and ED are perpendicular to BD, \angle ACB = 60^\circ, \angle CAE = 45^\circ, and \angle AEC = 45^\circ. If AB = \sqrt{3}, what is the perimeter of quadrilateral ABDE?

\textbf{(A)}\ 1+\sqrt{2}+2\sqrt{3}\quad \textbf{(B)}\ 2+2\sqrt{2}+\sqrt{3}\quad \textbf{(C)}\ 1+2\sqrt{2}+2\sqrt{3}\quad \textbf{(D)}\ 2+\sqrt{2}+2\sqrt{3}\quad \textbf{(E)}\ 2+2\sqrt{2}+2\sqrt{3}

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