6. Solve the system of equations: $\left\{\begin{array}{l}x^{3}+y^{3}=3 y+3 z+4, \\ y^{3}+z^{3}=3 z+3 x+4, \\ z^{3}+x^{3}=3 x+3 y+4 .\end{array}\right.$
(2014, German Mathematical Olympiad)
Hint: Subtract the equations from each other in pairs, first prove $x=y=z$.
The solution to the original system of equations is
$$
(x, y, z)=(-1,-1,-1),(2,2,2) .
$$