VTMO 2026 - Grade 9 - Final - SET 8 - Self-study Question 4

Find all real triples $(a,b,c)$ for which the following identity holds for every real $x,y,z$:
$$|ax+by+cz|+|ay+bz+cx|+|az+bx+cy|=|x|+|y|+|z|.$$