An electric vehicle manufacturing company builds two types of cars, an $x8$ and an A$3.$ Car manufacturing has three stages: $1,2,$ and $3.$X$8$ takes an hour in Stage $1,$ an hour in Stage $2,$ and $30$ minutes in Stage $3.$ On the other hand, A$3$ takes an hour and a half in Stage $1,45$ minutes in Stage $2,$ and another half an hour in Stage $3.$ The company must manufacture at least one car of each type per day to meet the market demand. However, each day, they have only seventeen and a half hours allocated for Stage $1,$ fourteen hours for Stage $2,$ and fifteen hours for Stage $3.$ If the profit for each X$8$ is $$2800$ and each A$3$ is $$3000,$ then find out how many of each type of car the manufacturer should produce daily to maximize the profit. Keep in mind that due to market limitations$,$ they only want to manufacture up to five of each type of car.