Find the smallest natural number $k$ such that the system of equations $$x+y+z=x^2+y^2+z^2=\dots=x^k+y^k+z^k $$has only one solution for positive real numbers $x$, $y$ and $z$.
Problem
Source: Kosovo Math Olympiad 2025, Grade 12, Problem 2
Tags: algebra, Kosovo, national olympiad, get the smallest, beautiful, Holder