Data collected at a parking lot indicate that a total of $310$ cars park between $8$:$00$ a$.$m$.$ and $5$:$00$ p$.$m$.$ Thirty$-$five percent of these cars are parked for an average of $2$ hr$,10\%$ for an average of $3$ hr$,10\%$ for an average of $4$ hr$,$ and the remaining cars are parked for an average of $9$ hr$.$ Ten percent of the parking bays are vacant on average $($between $8$ a$.$m$.$ and $5$ p$.$m$.)$ at the parking lot. The owner of the parking lot is planning an expansion of her lot to provide adequate demand for the following $10$ years. If she has estimated that parking demand for all categories will increase by $6\%$ a year, determine the number of additional parking bays that will be required. $($Round your answer to the nearest integer.$)$