Can we find the area of this inscribed quadrilateral?

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The diagram shows the ellipse whose equation is \(x^2+y^2-xy+x-4y=12\). The curve cuts the \(y\)-axis at points \(A\) and \(C\) and cuts the \(x\)-axis at points \(B\) and \(D\). What is the area of the inscribed quadrilateral \(ABCD\)?