A light beam is travelling from Region $I$ to Region $IV$ (Refer Figure). The refractive index in Regions $I, II, III$ and $IV$ are $n_0,\frac{n_0}{2},\frac{n_0}{6} \, \, and \, \, \frac{n_0}{8},$ respectively.The angle of incidence $\theta$ for which the beam just misses entering Region $IV$ is