\[ \sqrt{ \frac{2}{2^{2^0}} + \sqrt{ \frac{2}{2^{2^1}} + \sqrt{ \frac{2}{2^{2^2}} + \sqrt{ \frac{2}{2^{2^3}} +\cdots }}}} \]

If the value of above expression is in the form \(\sqrt{A}\), find the value of \(A\).

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