KöMaL 2025 A917

Let a set $S$ of complex numbers be called symmetric if the complex conjugate of every element of $S$ also belongs to $S$. For each positive integer $n$, determine the largest positive integer $K_n$ for which there exists an $n$-element symmetric set $S = \{z_1, z_2, \ldots, z_n\}$ not containing $0$, such that $z_1^k + z_2^k + \ldots + z_n^k \leq 0$ holds true for every integer $1 \leq k \leq K_n$.