The three little pigs received a loan for $$9000$ at an effective annual interest rate of $5\%$ from BBW Bank to make repairs on their brick house. They will repay the loan, starting one year after the loan was disbursed, by making annual payments equal to $5$ times the amount of interest due until they can make a final payment of no more than $$425.$ Determine how many payments they will make and the amount of the final payment.

$($a$)$ The three little pigs will make payments.

$($b$)$ The final payment will be for $