$4.$ A loan of amount L is to be repaid at an effective annual interest rate of $8\%$ with payments at the end of each year, starting one year after the loan. It can be repaid under the following three options :

$($i$)20$ equal annual payments of K each.

$($ii$)$ Installments of Y each year for $20$ years plus interest on the unpaid balance.

$($iii$)$ Annual payments of $($K $+200)$ each plus a final smaller payment of Z one year after the last regular payment $($where K is the same as in $($i$)).$

If the total amount of interest of $16800$ is paid on the loan under option $($ii$),$ calculate K and for option $($iii$)$ find the total number of payments and the amount Z