Evaluate $abcd$

Let $a, b, c, d$ be real numbers such that $\displaystyle a=\sqrt{4-\sqrt{5-a}}$, $\displaystyle b=\sqrt{4+\sqrt{5-b}}$, $\displaystyle c=\sqrt{4-\sqrt{5+c}}$ and $\displaystyle d=\sqrt{4+\sqrt{5+d}}$. Calculate $abcd$.

This is one very intriguing problem because I am sure your instinct will tell you, No, please don't multiply all the given four expression out! That is a surest way to get into a big mess if you do that, and there got to be an easy way out for this problem.