131. Let $x=1, y, z$ be any three numbers; $x_{1}$, $y_{1}, z_{1}$ be the absolute values $|x-y|,|y-z|,|z-x|$ of the pairwise differences of these numbers; $x_{2}, y_{2}, z_{2}$ be the absolute values of the pairwise differences of the numbers $x_{1}, y_{1}, z_{1}$ (i.e., the numbers $\left.\left|x_{1}-y_{1}\right|,\left|y_{1}-z_{1}\right|,\left|z_{1}-x_{1}\right|\right) ; x_{3}, y_{3}, z_{3}$ be the absolute values of the differences of the numbers $x_{2}, y_{2}, z_{2}$, and so on. It is known that for some $n$ the triplet of numbers $x_{n}, y_{n}, z_{n}$ does not differ from the triplet of numbers $x, y, z$. What are the values of the numbers $y$ and $z$?
131. First of all, let's determine for which numbers \(x, y, z\) the triples \((x_n, y_n, z_n)\) and \((x, y, z)\) can coincide. Since all triples of numbers starting from \(x_1, y_1, z_1\) are non-negative, the numbers \(x\), \(y\), and \(z\) must also be non-negative. Let's agree to consider that \(x \geqslant y \geqslant z\) and \(x_i \geqslant y_i \geqslant z_i\) for all \(i=1,2,3, \ldots\). Since, obviously, \(x_i = x_{i-1} - z_{i-1}\) for all \(i \geqslant 1\) (where \(x_0\) and \(z_0\) are understood to be the numbers \(x\) and \(z\)), it follows that \(x \geqslant x_1 \geqslant x_2 \geqslant x_3 \geqslant \ldots\), and if at least one number \(z_i (i=0,1,2, \ldots)\) is greater than zero, then \(x_{i+1}\) is also less than \(x\); therefore, if \(x_n = x\), then \(z_i = 0\) for \(i=0,1, \ldots, n-1\). Thus, it must be that \(z=0\) and \(z_1=0\), from which it follows that either \(y=x\) (and \(z_1 = x - y = 0\)), or \(y = z = 0\) (and \(z_1 = y - z = 0\)). Therefore, we see that for the triple of numbers \((x, y, z)\) to coincide with any triple \((x_n, y_n, z_n)\), it is necessary that the initial triple, when \(x=1\), either has the form \((1,1,0)\) or the form \((1,0,0)\). The second case can be immediately discarded, because from the triple \((1,0,0)\) we transition to the (different from the initial) triple \((1,1,0)\); therefore, if \(x=1\) and the triple \((x, y, z)\) coincides with \((x_n, y_n, z_n)\), then \((x, y, z)\) is the triple of numbers \((1,1,0)\) (and the triples \((x_n, y_n, z_n)\) for all \(n\) have the same structure):