<s>[INST] Solve the system ${x}^{2}+ \left( y-1 \right) ^{2}=4,z^{4}+y{z}^{2}+xz+1=0. $ Find all real solutions of  the system of equation
\begin{cases}
{x}^{2}+ \left( y-1 \right) ^{2}=4,\\{z}^{4}+y{z}^{2}+xz+1=0.
\end{cases}
 [/INST] let $t = y-1$
$x^2+t^2 = 4$
$tz^2 + xz = -z^4 - z^2 - 1$
Solutions (x,t) exist if and only if $\frac{z^4 +z^2 +1}{\sqrt{z^4 +z^2}} \le 2$ (geometrically)
Let $k = z^4 + z^2$
$\frac{k +1}{sqrt(k)} \le 2$ equals $k^2 +2k +1 \le 4k$  equals $k^2 -2k +1 \le 0$ equals k = 1;
So, $z^4 + z^2 = 1$
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