Nemotron-IMO-Bench 2026 A25

Let $n$ be a positive integer. Find the largest positive integer $K_n$ for which there is a set $S=\{z_1,z_2,\dots,z_n\}$ of $n$ complex numbers, stable under complex conjugation, not containing $0$ and such that $z_1^k+...+z_n^k\leq 0$ for all $1\leq k\leq K_n$.