Let
$A:=\sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}}$$B:=\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}}}}$
Find $B-A$.

Details and assumptions:

The sign pattern in the first radical is $(--+)$.

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