For the dishwasher inventory problem, Example 2, relabel the state space such that states \((0,0),(0,1)\), and \((0,2)\) are lumped together as, say, state \(0^{*}\), and states \((1,1)\) and \((1,2)\) are lumped together as state \(1^{*}\). Write the new transition matrices \(T_{a}\), and write dynamic programming equations similar to (6) for the redesigned problem. Expand the equations out in full.