Mark just graduated from college and has started his first job in sales. Based on conversations with his manager, Mark believes his income will increase substantially over the next couple of years. His current gross monthly income is $$3,450.$ Although Mark's current car works fine, it does have a lot of miles on it$.$ As a result, he would like to purchase a new car. Mark has some student loans, credit card debt, and a car loan. His monthly required payments are $$150$ for student loans, $$120$ for credit card debt, and $$220$ for his car loan. He has a roommate and currently pays $$645$ per month for his portion of the rent.

$($a$)$

If Mark wants to keep his debt$-$to$-$income ratio less than $30\%,$ what is the maximum monthly payment that he could have on a new car loan? Assume that Mark could sell his current car and pay off the remaining balance of his current car loan.

The maximum monthly payment that Mark could have on a new car loan is S