Evaluating $\lim\limits_{k\to\infty}\sum\limits_{r=1}^k\tan^{-1}\left(\frac{6^r}{2^{r+1}+3^{2r+1}}\right)$

Question

Let $$S_k=\sum_{r=1}^k\tan^{-1}\left(\frac{6^r}{2^{r+1}+3^{2r+1}}\right)$$ Then find the value of $$\lim_{k\to\infty}S_k$$

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I can clearly see that the value of the expression inside the $\tan^{-1}$ function lies in the interval $\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$. Also, I tried to apply the sum formula of $\arctan$ function but in vain. Any help is greatly appreciated.