$\sqrt{ 2^1+ \sqrt{ 2^2+ \sqrt{ 2^4 + \sqrt{2^8 + \cdots } } } }$

The value of above expression is in the form $\phi \sqrt{A}$, where $A$ is an integer. Find $A$.

**Notation**: The symbol $\phi$ denotes the golden ratio, $\phi = \dfrac{1+\sqrt5}2$.

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