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 $abcd$.
abcd = 47.
