Example 1 Solve the system of equations:
$$
\left\{\begin{array}{l}
x+y-z=-1, \\
x^{2}-y^{2}+z^{2}=1, \\
-x^{3}+y^{3}+z^{3}=-1 .
\end{array}\right.
$$
(52nd Dutch Mathematical Olympiad)
【Analysis】From equation (1), it is easy to think of substitution and elimination. Combining with equation (2), after elimination, the expression can still be factored, thus solving the problem.
Solution From equation (1), we get $z=x+y+1$.
Substituting into equation (2) and rearranging, we get
$$
2(x+y)(x+1)=0 \text {. }
$$
(1) If $x+y=0$, then $y=-x, z=1$.

Substituting into equation (3) yields
$$
(x, y, z)=(1,-1,1) \text {. }
$$
(2) If $x=-1$, then $z=y$.

Substituting into equation (3) yields
$$
(x, y, z)=(-1,-1,-1) \text {. }
$$

In summary, $(x, y, z)=(1,-1,1),(-1,-1,-1)$.