IMO selected problem

Geometry Level 4

\[F(x,y)=\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x-1)^2+(y+1)^2}+\sqrt{(x+2)^2+(y+2)^2}\] Let the minimum of the expression above is \(S\). If \(S^{2}=k+m\sqrt{n}\), where \(n\) is square free, find \(m+k+n\).

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