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If angle $\alpha$ has a tangent of $\cfrac{{12}}{5}$, and lies in Quadrant III, and angle $\beta$ has a secant of $\cfrac{8}{3}$ and lies in Quadrant IV, determine $\sec{(\alpha+\beta)}$.

A

$\cfrac{520-416\sqrt{55}}{2565}$

B

$\cfrac{346\sqrt{55}-13}{2565}$

C

$\cfrac{-15-12\sqrt{55}}{104}$

D

$\cfrac{-15-12\sqrt{55}}{-104}$

E

$-\cfrac{1}{104}$

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