A casino is considering adding a new game to their collection, but need to analyze it before releasing it on their floor. They have hired you to execute the analysis. On each round of the game, the player has the option of rolling a fair $6-$sided die. That is$,$ the die lands on values $1$ through $6$ with equal probability. Each roll costs $1$ dollar, and the player must roll the very first round. Each time the player rolls the die, the player has two possible actions:

$1.$ Stop: Stop playing by collecting the dollar value that the die lands on$,$ or $2.$ Roll: Roll again, paying another $1$ dollar.

You decide to model this problem using an infinite horizon Markov Decision Process $($MDP$).$ The player initially starts in state Start, where the player only has one possible action: Roll. State si denotes the state where the die lands on i$.$ Once a player decides to Stop, the game is over, transitioning the player to the End state.

$($a$)$ In solving this problem, you consider using policy iteration. Your initial policy $\backslash$pi is in the table below. Evaluate the policy at each state, with $\backslash$gamma $=1.$