Rectangle in an Ellipse

Geometry Level 3

\(A,B,C,D\) are consecutive vertices of a rectangle whose area is \(2006\) square units. An ellipse with area \(2006\pi\), which passes through \(A\) and \(C\) has its foci at \(B\) and \(D\).

If the perimeter of the rectangle can be expressed as \(a\sqrt{b}\) where \(a\) is a positive integer and \(b\) is a square-free positive integer, find \(a+b\).

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