23. Find all non-zero integer triples $\{a, b, c\}$, satisfying the conditions:
$$a \equiv b(\bmod |c|), \quad b \equiv c(\bmod |a|), \quad c \equiv a(\bmod |b|)$$
23. Similar to the previous problem, we can first assume $(a, b, c)=1, c>0$ and $|a| \leqslant|b| \leqslant c$. It follows that we must have (i) $a=b$; (ii) $a-b= \pm c$; (iii) $a-b= \pm 2 c$. These three cases can respectively yield (i) $\{1,1, c\}$, $\{-1,-1, c\}$; (ii) $\{1,-n, n+1\},\{2,-(2 n+1), 2 n+3\}, n$ being any positive integer, and $\{-1,1,2\},\{-1,2,3\}$; (iii) $\{1,-1,1\},\{-1,1,1\}$.