A reservoir holds 3 units of water. We will control the chain defined by \(X_{n}=\#\) units of water in the reservoir at the beginning of month \(n\), by deciding how much water to release from the reservoir at the beginning of the month. Each unit of water released produces a monetary benefit of 1 unit, due to the production of power and irrigation. But if the reservoir is dry, there is a loss of 2 monetary units due to the need to purchase power from another source. During a month, rainfall produces either 0 units or 1 unit of inflow to the reservoir, each with probability \(1 / 2\). Any rainfall occurring when the reservoir is already full is simply lost. Use the method of successive approximations to find the optimal value function and optimal water release policy for the infinite horizon Markov decision problem with discount factor \(\alpha=.95\).