Solve a Small Bellman System by Hand

Consider a 2-state simplification of the notification MDP: states OK and Fatigued, absorbing Churned state with value 0. Under the fixed policy "always send," \gamma = 0.8:

1. Write the Bellman expectation equations for V(\text{OK}) and V(\text{Fatigued}) and solve the resulting linear system.
2. Suppose instead the agent could wait in Fatigued, with reward -0.05 and transitions \{OK: 0.5, Fatigued: 0.4, Churned: 0.1\}. Using your V(\text{OK}) from part 1, compute Q(\text{Fatigued}, \text{wait}) and compare it to Q(\text{Fatigued}, \text{send}) implied by part 1. What does policy improvement say the agent should do in Fatigued?
3. State, without recomputing, whether this new value can ever be lower than under the "always send" policy, and why.