A Problem in Complex Numbers!

Algebra Level 4

Let $z_1,z_2,z_3$ be 3 complex numbers lying on a circle whose centre is at the origin such that $z_i+z_{j}z_k$ (where $i,j,k\in \{1,2,3\}$ and $i \ne j \ne k$ ) are real numbers, then find $z_1\times z_2\times z_3$ .

Note

$z_1\ne z_2\ne z_3$ . In other words, $z_1,z_2,z_3$ are all distinct points on the circle.

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