If \( \displaystyle \sum_{r=1}^{13} \left\{ \frac{1}{\displaystyle \sin\!\left( \frac{\pi}{4} + (r-1)\frac{\pi}{6} \right)\,\sin\!\left( \frac{\pi}{4} + \frac{r\pi}{6} \right)} \right\} = a\sqrt{3} + b,\; a,b\in\mathbb{Z} \), then \( a^{2}+b^{2} \) is equal to:
