Define $f(x)=\frac{1+x}{1-x}$ and $g(x)=\frac{-2}{x+1}$. Find the value of \[g(f(g(f(\dotsb g(f(12)) \dotsb ))))\] where the function $f$ is applied 8 times, and the function $g$ is applied 8 times, alternating between the two.
(A)12 (B)8 (C)4 (D)0
C