Find all complex numbers $a_i$ such that $\prod\limits_{j\neq i=1}^k \ (a_i-a_j)^n=1$

Question

$n>1$ is fixed.
Find all complex numbers $a_1,\dots,a_k\;$ such that $\prod\limits_{j\neq i=1}^k \ (a_i-a_j)^n=1.$
What I did until now was solving for $k=3,2$ that it was not hard.