$(25$ points$)$ A professor has been contacted by three not$-$for$-$profit agencies that are willing to work with student consulting teams. The agencies need help with such things as budgeting, information systems, and coordinating volunteers. Although each of the three student teams could work with any of the agencies, the professor feels that there is a difference in the amount of time it would take each group to solve each problem. The professor's estimate of the time, in days, is given in the table below. Develop and solve a linear programing model to see which team works with which project. At least how many days are needed to finish the project?

$\backslash$table$[[,$Projects$],[$Team$,$Budgeting,Information,Volunteers$],[$A$,32,35,21],[$B$,38,40,19],[$C$,41,42,25]]$