Pig is a $2-$player game played with a $6-$sided die. On your turn, you can decide either to roll the die or to pass. If you roll the die and get a $1,$ your turn immediately ends and you get $1$ point. If you instead get some other number, it gets added to a running total and your turn continues $($i$.$e$.$ you can again decide whether to roll or pass$).$ If you pass, then you get either $1$ point or the running total number of points, whichever is larger, and it becomes your opponent$$s turn. For example, if you roll $3,4,1$ you get only $1$ point, but if you roll $3,4,2$ and then decide to pass you get $9$ points. The first player to get to $100$ points wins. Suppose at some point the player whose turn it is has x points, their opponent has y points, and the running total for this turn so far is z$.$ Let$$s work out what the optimal strategy is$.4.$A$.(5$ pts$.)$ Let W$($x$,$ y$,$ z$)$ be the probability that the current player will eventually win if both players play optimally. What is W$($x$,$ y$,$ z$)$ if x $+$ z $100?4.$B$.(5$ pts$.)$ Suppose the current player decides to roll, and gets a $1.$ What$$s the probability that they$$ll still win, in terms of the function W$?4.$C$.(5$ pts$.)$ Give a recursive formula for W$($x$,$ y$,$ z$).4.$D$.(5$ pts$.)$ Describe a dynamic programming algorithm to compute W$($x$,$ y$,$ z$).$ If you needed N points to win instead of $100,$ what would be the asymptotic runtime of your algorithm?