Let z be a complex number such that $z^2 +z + 1/z^2 + 1/z + 1=0 $ If n is a natural number then find the value of  $ z^{2012n} + z^{1006n} + 1/z^{2012n} +1/z^{1006n} $ is equal to.
I tried rewriting it as $ t^2+t-1=0 $ where $ t=z+1/z $ and then find roots but I don't know how to use it to get required value.

HINT:
As $z\ne0,$  multiply throughout by $z^2$
$$z^4+z^3+z^2+z+1=0\implies z^5-1=(z-1)(z^4+z^3+z^2+z+1)=0\implies z=e^{2\pi m i/5}$$ where $m\equiv1,2,3,4\pmod5$
