Orthogonal parabolas

Algebra Level 4

The two orthogonal parabolas \[\begin{cases} y=(x+1)^2 \\ (y-2.5)^2=x+4 \end{cases}\] intersect at four distinct points, \(A=(x_1,y_1), B=(x_2, y_2) \), \( C=(x_3, y_3)\) and \(D=(x_4, y_4)\). Let \(S=x_1+x_2+x_3+x_4\) and \(R=y_1+y_2+y_3+y_4\). Find the value of \(\lfloor{1000(R+S)}\rfloor\).

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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