Find All 17 real solutions $(w,x,y,z)$ to the following system of equalities:
$$
2w+w^2x=x
$$
$$
2x+x^2y=y
$$
$$
2y+y^2z=z
$$
$$
-2z+z^2w=w
$$

I was thinking of using something related to cyclic sums, but I couldn't figure it out since one of the signs is negative, so I get 
$$
\sum\_{cyc} (x+x^2y)=4z
$$
so finding the zeroes is no longer an easy process.

I also tried to find upper and lower bounds for the cyclic sum in the left hand side, using the AM-GM inequality but that really didn't help.

So I don't really know what to do, other than solving for each variable and substituting to end up with a monstrous polynomial, which makes me think there is an easier method I'm overlooking