8- 22 Given 3 numbers $x, y, z$. The absolute values of their pairwise differences are denoted as $\dot{x}_{1}, y_{1}, z_{1}$. The absolute values of the pairwise differences of $x_{1}, y_{1}, z_{1}$ are denoted as $x_{2}, y_{2}, z_{2}$. Continuing this process, after $n$ steps, it is found that $x_{n}=x, y_{n}=y, z_{n}=z$. If it is known that $z=1$, try to find $x$ and $y$.
[Solution] Clearly, $x=x_{n} \geqslant 0, y=y_{n} \geqslant 0, \approx=z_{n} \geqslant 0$. For any two non-negative real numbers $a, b$, it is easy to see that
$$|a-b| \leqslant \max (a, b)$$

and the equality holds if and only if at least one of $a, b$ is 0. Therefore, at least one of $x, y, \approx$ is 0, and at least one of $x_{k}, y_{k}, z_{k}(1 \leqslant k \leqslant n)$ is 0. Thus, two of $x, y, z$ must be equal. Based on the above analysis, a simple verification yields
$$(x, y)=(0,0) \text { or }(0,1) \text { or }(1,0) .$$