If $\cot \alpha=1$ and $\sec \beta=-\frac{5}{3}$, where $\pi<\alpha<\frac{3 \pi}{2}$ and $\frac{\pi}{2}<\beta<\pi$, then the value of $\tan (\alpha+\beta)$ and the quadrant in which $\alpha+\beta$ lies, respectively are
Select the correct option:
A
$-\frac{1}{7}$ and $\mathrm{IV} \mathrm{I}^{\text {th }}$ quadrant
B
7 and $\mathrm{I}^{\text {st }}$ quadrant
C
-7 and $\mathrm{IV}^{\text {th }}$ quadrant
D
$\frac{1}{7}$ and $\mathrm{st}^{\text {st }}$ quadrant
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