The bumper car section of an amusement park is a popular attraction that generates a significant portion of the park$$s revenue. However, from time to time, cars break down and need to be repaired. This not only affects the park$$s reputation but also leads to a loss of revenue as customers are unable to use the attraction. The manager of the amusement park is faced with the task of hiring a repair team to fix the broken cars. The average number of cars that break down in the bumper car section is $2$ every hour. The loss of revenue due to a breakdown of each car is $$40$ per hour. The manager can choose from two types of repairmans to hire: For the first type of repairman, the average repair time of each car is $20$ minutes, and for the second type, it is $15$ minutes. Hiring the first type repairman costs $$40$ per hour, but hiring the second type$$s cost is $$60$ per hour as the repairmans of this type have higher expertise and can finish repairing each car faster. Considering the above information, answer the following questions: $($a$)$ Assume that the manager just wants to hire a single repairman. Determine the type of the repairman that the manager should choose to hire. $[$Hint: Calculate the total hourly cost of each type $($cost of hiring the repairman and the cost of lost revenue due to car breakdowns$),$ and choose the option with a lower cost.$]($b$)$ Now consider the type chosen in the previous part. Determine the minimum number of that type of repairmans which should be hired to keep the expected breakdown time of each car $($time of waiting to be repaired and repair time itself$)$ lower than $20$ minutes. $($Provide the numerical answers and show all the calculations$)$