Powerful Product

Calculus Level 5

$\large \prod _{n=0}^\infty\left(\frac1{2^{2^{n+1}}}-\frac1{2^{2^n}}+1\right)$

The above infinite product can be written as $\frac AB$ , where $A$ and $B$ are coprime positive integers. What is the value of $A+B?$