3. Find a set of real numbers $(x, y, z)$ (where $x$, $y$, and $z$ are distinct) that satisfies
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=-x+3 y+z, \\
y^{2}+z^{2}=x+3 y-z, \\
x^{2}+z^{2}=2 x+2 y-z .
\end{array}\right.
$$
3. (1)-(2) and rearranging gives
$$
\begin{array}{l}
x^{2}-z^{2}=-2(x-z) \\
\Rightarrow(x-z)(x+z+2)=0 \\
\Rightarrow z=-x-2 .
\end{array}
$$

Similarly, (3)-(2) and rearranging gives $y=1-x$.
Substituting $z=-x-2, y=1-x$ into equation (1) yields
$$
\begin{array}{l}
x^{2}+(1-x)^{2}=-x+3(1-x)+(-x-2) \\
\Rightarrow 2 x^{2}+3 x=0 \\
\Rightarrow x=0, x=-\frac{3}{2} .
\end{array}
$$

Therefore, $(x, y, z)=(0,1,-2),\left(-\frac{3}{2}, \frac{5}{2},-\frac{1}{2}\right)$.