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If $\sqrt{\sqrt[3]{(a+1)^{5}}}=(a+1)^{b}$, then find the value of $b$.

A

$2$

B

$\cfrac{1}{3}$

C

$\cfrac{2}{3}$

D

$\cfrac{2}{5}$

E

$\cfrac{5}{6}$

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