Minimizing $x^2+y^2+z^2$ subject to $xy -z + 1 = 0$ via Lagrange multipliers $$\begin{array}{ll} \text{minimize} & f(x,y,z) := x^2 + y^2 + z^2\\ \text{subject to} & g(x,y,z) := xy - z + 1 = 0\end{array}$$

I tried the Lagrange multipliers method and the system resulted from has no solution. So I posted it to see if the question is wrong by itself or I'm missing something.
So I made the Lagrangian equation $L(x,y,z,λ)=x^2 + y^2 + z^2 + λ(xy -z+1)$
and then
$θL/θx = 2x + λy =0$
$θL/θy = 2y + λx =0$
$θL/θz = 2z - λ =0$
$θL/θλ = xy -z +1 =0 $
The obvious solution for that system is x=0 , y=0 , z=1 and λ=2
But solving it in an online solver for nonlinear systems of equation the answer I get is that it's unsolvable.
So my question is: What I'm doing wrong

$$x^2+y^2+z^2=x^2+y^2+z^2+2(xy-z+1)=(x+y)^2+(z-1)^2+1\geq1.$$
The equality occurs for $x=y=0$ and $z=1$, which says that we got a minimal value.
