$1.$ Twiggy is formulating plans to extend its services to new Tier$-$II cities as part of its expansion strategy. The company predicts that the demand for orders in this new city will adhere to a normal distribution, with a mean of $1000$ and a standard deviation of $200.$ Several restaurants can be highlighted on the Twiggy app. The estimated average cost for restaurants to prepare a meal is Rs$.50$ per order. Twiggy aims to charge customers an average value of Rs$.200$ per order via its app. Each restaurant is expected to manage approximately $70$ orders per day. Twiggy has committed to compensating any shortage in orders falling below this threshold at a rate of Rs$.30$ per order. This approach will enable Twiggy to guarantee restaurants a minimum number of orders and foster trust. Moreover, Twiggy has decided to remunerate restaurants Rs$.100$ per order, in addition to $25\%$ of the revenue it accumulates from each order.

a$.$ Based on the data provided, how many restaurants should Twiggy aim to enlist on its app? Please round up the number of restaurants to the nearest whole number.

b$.$ What is the expected profit for Twiggy based on the answer of $($a$)?$

c$.$ If a restaurant spends Rs$.50$ on preparing each order, any unused material at the end of the day will have to be discarded. Each restaurant is prepared to deliver $70$ orders. If the expected sales from the Swiggy app are evenly distributed across all participating restaurants, what would be the expected profit for an individual restaurant?