A set of four distinct real numbers is called round if it satisfies $\begin{cases}
x_1^2+x_2=2,\\
x_2^2+x_3=2,\\
x_3^2+x_4=2,\\
x_4^2+x_1=2,\\
\end{cases}$
and $x_1 = \min (x_1, x_2, x_3, x_4 )$.

How many round sets are there?

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