Consider the complex numbers $z_1, z_2, \cdots , z_n$ with $|z_1| = |z_2| = \cdots = |z_n| = r > 0$. Prove that the number $$E =\frac{(z_1 + z_2)(z_2 + z_3)\cdots(z_{n-1} + z_n)(z_n + z_1)}{z_1z_2 \cdots z_n}$$ is real.