GARCH term structure after a shock

Illustrative Nifty GARCH(1,1): $\alpha = 0.10$, $\beta = 0.85$, long-run volatility 16% a year (252 days). After a sell-off, $\sigma_{T+1}$ is 30% annualised. (a) Find $\omega$ and the half-life. (b) Forecast the volatility (annualised) for day 10, and the average volatility over the next 21 trading days. (c) A 1-month Nifty option is trading at an implied volatility of 24%. Compare and give two reasons the two numbers can legitimately differ.