$2.$ You have borrowed a loan form a bank for $$500,000$ to finance a new house. You are required to repay the loan in $360$ equal monthly payments starting at the end of month $1.$ The bank charges an interest rate of $8\%$ APR, compounded monthly. You have already paid $120$ payments in the current plan. The interest dropped to $6\%$ and the bank has offered to refinance at the lower rate. The expectation is that your monthly payment will be lowered due to refinancing. The bank charges $3\%$ of the remaining balance as the service fee. Find the following:

$($i$)$How soon you can repay the loan if you decide to pay additional $10\%$ payment each month?$($ Relates to the original plan of $30-$year loan at $8\%$ APR$).$

$($ii$)$ Would you refinance the loan at the new $(6\%)$ interest rate? Explain.

$($iii$)$ What $\%$ of your first mortgage payment is used towards paying the interest? Where does the rest amount go$?$