Let $a$, $b$, $c$, and $d$ be distinct real numbers such that \begin{align*} a & = \sqrt{4 + \sqrt{5 + a}}, \\ b & = \sqrt{4 - \sqrt{5 + b}}, \\ c & = \sqrt{4 + \sqrt{5 - c}}, \\ d & = \sqrt{4 - \sqrt{5 - d}}. \end{align*}compute $abcd$.