Let a, b, c, and d be positive real numbers satisfying a b c d=1. Prove that
\frac{1}{\sqrt{\frac{1}{2}+a+a b+a b c}}+\frac{1}{\sqrt{\frac{1}{2}+b+b c+b c d}}+\frac{1}{\sqrt{\frac{1}{2}+c+c d+c d a}}+\frac{1}{\sqrt{\frac{1}{2}+d+d a+d a b}} \geq \sqrt{2}