A company provides three types of services, and the profit from one unit of each service is $5,4,$ and $2,$ respectively. Two resources are used to provide services. Resource $1,$ which can be used by a total of $31$ people, requires $6,4,$ and $2,$ and Resource $2,$ which can use a total of $40$ people, requires $7,13,$ and $5$ to provide each service. let's do it$.$ When calculating the amount of each service to maximize the total profit through service provision, what is the amount of service $3$ provided? $($Round to $3$ decimal places.$)$