Dominated 2! Its Algebra cum Geometry

$\sqrt{\frac{1}{2}}\times \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}\times \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}}\times \cdots = \dfrac{a}{b\pi }$

The equation above holds true for coprime positive integers $a$ and $b$ , find the value of $a^2 + 3b$ .