# Shivam, the Summation King

**Geometry**Level 4

\[\large \sum_{m=1}^{6}\csc\left(\omega+\frac{(m-1)\pi}{4}\right)\csc\left(\omega+\frac{m\pi}{4}\right)=4\sqrt{2}\] There exists some angle \(\omega\) with \(0<\omega<\frac{\pi}{3}\) such that the above equation is satisfied. Find \[\sum_{r=1}^{7}\tan^{2}(\frac{3r\omega}{4})\]