Example 7 Solve the system of equations: $\left\{\begin{array}{l}x^{4}+y^{2}+4=5 y z, \\ y^{4}+z^{2}+4=5 z x, \\ z^{4}+x^{2}+4=5 x y .\end{array}\right.$ $[6]$
(61st Czech and Slovak Mathematical Olympiad)
【Analysis】From the terms and their frequencies, we can consider completing the square, or we can use the mean inequality.
Solution Adding the three equations yields
$$
\begin{array}{l} 
x^{4}+y^{4}+z^{4}+x^{2}+y^{2}+z^{2}+12-5 x y-5 y z-5 x z=0 \\
\Rightarrow\left(x^{2}-2\right)^{2}+\left(y^{2}-2\right)^{2}+\left(z^{2}-2\right)^{2}+ \\
\frac{5}{2}\left((x-y)^{2}+(y-z)^{2}+(z-x)^{2}\right)=0 \\
\Rightarrow x=y=z \\
\Rightarrow(x, y, z)=( \pm \sqrt{2}, \pm \sqrt{2}, \pm \sqrt{2}) .
\end{array}
$$