Area from a Perpendicular Sum

Geometry Level 3

\(ABCD\) is a convex cyclic quadrilateral such that \( AB = AD\) and \( \angle BAD = 90^\circ\). \(E\) is the foot of the perpendicular from \(A\) to \(BC\), and \(F\) is the foot of the perpendicular from \( A\) to \(DC\). If \( AE + AF = 16\), what is the area of \(ABCD\)?

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