The two orthogonal parabolas
\[\begin{cases} y=2x^2+a \\ x=2y^2+b \end{cases}\]
intersect at four distinct points. These four points on a same circle of area 10. Given that \(a, b<0\), find the maximum value of \(\lfloor{1000ab}\rfloor\).

**Remark**: If the orthogonal parabolas intersect at four distinct points, then these four points are **always** on a same circle.

This problem is part of Curves... cut or touch?

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