Consider all polynomials of a complex variable, $P(z)=2z^3+a{z}^2+bz+c$, where $a,b,c$ are integers, $0\le c\le b\le a\le 2$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties?