Let $g(z) = \frac{a_1z + b_1}{c_1z + d_1}$ with $a_1,b_1,c_1,d_1 \in \mathbf{Z}$, $(a_1-d_1)^2 + 4b_1c_1 \ne 0$, and at least one of $c_1,d_1$ is non-zero. Now suppose that $g^n(z) = z$ and that $g^j(z) \ne z$ for $0<~$$j<n$ and define $\textrm{sgn}(g(z)) = n$. Find all possible values of $\textrm{sgn}(g(z))$.