Let $\omega$ be the complex number $\cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3}$. Then the number of distinct complex numbers $z$ satisfying $\begin{vmatrix} z+1 & \omega & \omega^{2} \\ \omega & z+\omega^{2} & 1 \\ \omega^{2} & 1 & z+\omega \end{vmatrix}=0$ is equal to

$\omega = e^{2\pi i/3},\quad \omega^{3}=1$

$1+\omega+\omega^{2}=0$

$\text{Find the number of distinct } z \text{ with}$

z+1 ω ω² ω z+ω² 1 ω² 1 z+ω = 0

$\text{1. Write } M = zI + A$

$\text{2. Study } A \text{ using } 1+\omega+\omega^{2}=0,\ \omega^{3}=1$

$\text{3. Reduce } \det(zI+A) \text{ and count distinct } z$

M = zI + A, A = 1 ω ω² ω ω² 1 ω² 1 ω

$\text{Row sums } = 1+\omega+\omega^{2}=0 \Rightarrow \det A = 0$

$\operatorname{tr}A = 1+\omega^{2}+\omega = 0$

$A^{2} = 0$

$\det(zI+A) = z^{3} + (\operatorname{tr}A)z^{2} + c_{2}z + \det A$

$\operatorname{tr}A = \det A = 0,\quad c_{2} = 0 \Rightarrow \det(zI+A) = z^{3}$

$z^{3}=0 \Rightarrow z=0\ \text{(multiplicity } 3\text{)}$

$\boxed{\text{Answer: } 1}$