JEE Math Practice Question

Let $$S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \pi}{6}\right)=-\frac{8}{\sqrt{3}}\right\}$$. Then

1. A.$$ S=\left\{\frac{\pi}{12}\right\} $$
2. B.$$ S=\left\{\frac{2 \pi}{3}\right\} $$
3. C.$$ \sum\limits_{\theta \in S} \theta=\frac{\pi}{2} $$
4. D.$$ \sum\limits_{\theta \in S} \theta=\frac{3\pi}{4} $$