A couple wants to purchase a $$260,000$ house, and they have the required $20\%$ down payment and money for other closing costs. The bank is offering a $30-$year mortgage at $4\mathrm{.}625\%$ interest compounded monthly. The couple has annual after$-$tax income of $$55,000,$ and other debts totaling $$650$ per month. $($a$)$ If the maximum debt$-$to$-$income ratio $($total monthly debt divided by after$-$tax monthly income$)$ is $43\%,$ can the couple afford to purchase the home? $($b$)$ If the couple lives in the house for $30$ years, what is the total amount paid for the house including down payment, principal, and interest?