$\displaystyle \text{Given }x=cy+bz,y=az+cx,z=bx+ay\text{ where } x,y,z$

$\displaystyle \text{ are not all zero, prove that }a^2+b^2+c^2+2abc=1. $

Our professor hinted that the brute force way of solving this is naive and that there is a simpler way, the problem is I don't see it. My idea was to set this into a matrix with 3 equations and 3 unknowns and try to solve x, y, and z. This approach started to become very messy and I would get an answer like $\displaystyle y=\frac{1-b^2}{a+cb}z$. So my question is really this, is there anybody out there that sees the trick for this problem that could give me a pointer. thanks in advance!