Mortgage is taken out by Nick Klonowski to buy a new house for $$400,000$ over $30$ years. Nick makes no down payment, borrowing the full amountof $$400,000$ at a fixed rate of$\mathrm{ 8}$ percent compounded monthly. Payments are made at the end of the month starting at the end of the first month.

$($a$)$What is the monthly payment?

$($b$)$What is the future value of the corresponding payments in $30$ years?

$($c$)$If the loan fees were embedded in the payment installments so that at a rate of $7\mathrm{.}8\%$ for the mortgage you still pay an APR of $8\%,$ what are the loan fees in today$$s dollars?