The Yankee Clipper Barber Shop employs five $(5)$ barbers and pays each a base rate of $$500$ per month. One of the barbers serves as the manager and receives an extra $$300$ per month. In addition to the base rate, each barber also receives a commission of $$3$ per haircut. A barber can do as many as $20$ haircuts a day. The Conner Barber Shop is a corporation with a $30\%$ tax rate and is open $24$ days a month.Other costs are incurred as follows: Advertising$$200$ per monthRent$$400$ per monthBarber Supplies$$0\mathrm{.}90$ per haircutUtilities$$175$ per month plus $$0\mathrm{.}35$ per haircutMagazines$$25$ per monthCleaning Supplies$$0\mathrm{.}15$ per haircutConner currently charges $$8$ per haircut.

Required:$($a$)$ Conner wants to earn $$2,160$ a month in after$-$tax operating profits. Compute the number of haircuts that must be given to reach this goal for the month of July, $2024.$