An automatic rain-detection system has three states: Green (low risk), Yellow (moderate risk), and Red (high risk). Each hour, the system transitions between states. If it is Green, there is a $0.7$ probability it stays Green, and $0.3$ probability it becomes Yellow. If it is Yellow, there is a $0.4$ probability it becomes Green, $0.4$ probability it stays Yellow, and $0.2$ probability it becomes Red. If it is Red, it stays Red with probability $1.0$. If the system starts in the Green state, what is the probability that it is in the Red state at the end of exactly 3 hours?