\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 $abcd$.

I set up each as a quartic and got $a^4-8a^2-a+11=0$ and similar equations for $b,c,d$, but am unable to figure out how to format the roots as a product of all of them. Could someone give me a bit of a hint? Thank you.