Preimage of a point under the Hopf map

Question

Considering the Hopf map $S^3 \longrightarrow S^2$ given by $$(x,y,z,w) \mapsto \bigl(x^2+y^2-z^2-w^2,2(xw+yz),2(yw-xz)\bigr),$$ I know that the preimage of point is a great circle in $S^3$. For example, the preimage of $(1,0,0)$ is the set $$\left\{(x,y,0,0) \in S^3\right\}.$$ I am trying to get the formula for a general point; taking a point $(X,Y,Z)$ in $S^2$, we have the following system of non-linear equations: $$\begin{cases} x^2+y^2+z^2+w^2=1, \\ x^2+y^2-z^2-w^2=X, \\ 2(xw+yz)=Y, \\ 2(yw-xz)=Z. \end{cases}$$ I don't know how to solve this system for $x, y, z$ and $w$.