HMMT 2006 U20

Compute the number of real solutions $(x, y, z, w)$ to the system of equations:

$\begin{array}{rlrl} x & =z+w+z w x & z & =x+y+x y z \\ y & =w+x+w x y & w & =y+z+y z w \end{array}$

Dataset text NuminaMath olympiads. Quoted with source.

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