ILL 1983 Problem 53

Let $a \in \mathbb R$ and let $z_1, z_2, \ldots, z_n$ be complex numbers of modulus $1$ satisfying the relation $$\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}$$ Prove that $a \in \{0, 1,\ldots, n \}$ and $z_k \in \{1, i \}$ for all $k.$