Powerful Product

Calculus Level 4

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

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

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