The same copy of this question is montioned here in SE with no convinced Answer , I want to know what MO will say about the below nested radical as a power form

it is well known that

$$\frac{2}{\pi}=\sqrt{\frac12}{\sqrt{\frac12+\frac12\sqrt{\frac12}}{\sqrt{\frac12+\frac12\sqrt{\frac12+\frac12\sqrt{\frac12}}}{\sqrt{\frac12+\frac12\sqrt{\frac12\cdots}}}}}$$

My Idea is to know what about above product if it is a power as shown below : $A=\sqrt{\frac12}^{\sqrt{\frac12+\frac12\sqrt{\frac12}}^{\sqrt{\frac12+\frac12\sqrt{\frac12+\frac12\sqrt{\frac12}}}^{\sqrt{\frac12+\frac12\sqrt{\frac12\cdots}}}}}$ ?