\[\begin{cases} x^2+y^2=1 \\ x^2-2xy+3y^2 =k\end{cases} \]

The above two curves touch each other at \(k=a\) and \(k=b\) where \(a>b\). Find the value of \(\left \lfloor{\dfrac{1000a}{b}}\right\rfloor\).

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