They are collinear, because you essentially introduced a unit root so that the steady state of these equations is not uniquely determined anymore within the model. Take the first equation: any value you pick as the steady state of A_N will be a steady state of this equation. Say A_N in steady state is 10:
10 = 10^rho_A_N*STEADY_STATE(10)^(1-rho_A_N)*1=10; This works for any value.
Usually, you fix
STEADY_STATE(A_N)^(1-rho_A_N)to set the steady state of A_N. But now you want to determine the steady state of A_N endogenously. This is simply not possible, because the equation does not contain enough information anymore.
This problem regularly occurs in New Keynesian models where the inflation rate and the nominal interest rate cannot be simultaneously endogenously determined. See [An infinity of steady states with Taylor rules)