$1.(20$ points$)$ A chocolatier must decide each week how many artisanal chocolates to make during the week to sell at Saturday's farmer's market. $-$ Each chocolate can be sold for $\backslash (\backslash$$ $20\backslash ).-$ He prefers to store no more than $7$ chocolates at his shop at any point in time. $-$ Each week, there is: $-\backslash (10\backslash \%\backslash )$ chance one customer wants to buy a chocolate, $-\backslash (10\backslash \%\backslash )$ chance he sees a demand of $2$ chocolates, $-\backslash (10\backslash \%\backslash )$ chance he sees a demand of $3$ chocolates, $-\backslash (10\backslash \%\backslash )$ chance he sees a demand of $4$ chocolate, $-\backslash (10\backslash \%\backslash )$ chance he sees a demand of $5$ chocolates, $-\backslash (20\backslash \%\backslash )$ chance he sees a demand of $6$ chocolates, $-\backslash (20\backslash \%\backslash )$ chance he sees a demand of $7$ chocolates, and $-\backslash (10\backslash \%\backslash )$ chance no customers are interested in buying chocolates. $-$ If he has no chocolates with him when a customer comes on Saturday, the customer leaves without buying one. $-$ On those weeks that he chooses to make chocolate, the chocolatier must use the kitchen that cost $\backslash (\backslash$$ $20\backslash )$ per week. $-$ In addition, it costs him $\backslash (\backslash$$ $2\backslash )$ to make each chocolate. Chocolates not sold at the end of the year can be sold for $\backslash (\backslash$$ $5\backslash )$ by then. Formulate this problem as a Markov decision process, in which the objective is to maximize the total expected income over the next $8$ weeks $($assuming there are only $8$ weeks left this year$).$ Clearly indicate the $8$ basic components of this MDP$.$