MathNet 2020 — Math Olympiad Vault — Part 05 Question 3

Let $|z_1|=|z_2|=|z_3|=1$ and $z_1z_2z_3\ne-1$. Prove that the following is real:
$$w=\frac{z_1+z_2+z_3+z_1z_2+z_1z_3+z_2z_3}{1+z_1z_2z_3}.$$