CN-TST 2022 Problem 5

Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions: (1) For any open arc $\Gamma$ of length $\pi$ on $C$ , there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$ . (2) For any open arc $\gamma$ of length $\pi/3$ on $C$ , there are at most $120$ of $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$ .

Find the maximum of $|z_1+z_2+\ldots+z_{240}|$ .