Please calculated the answer in a correct, precise, and comprehensive manner along with relevant formulas. Thanks

$$An amortized loan is a loan with scheduled periodic payments that are applied to both principal and interest. An amortized loan payment first pays off the relevant interest expense for the period, after which the remainder of the payment reduces the principal.

$($a$)$ A loan is created with $10$ annual equal payments of RM$500$ at an effective annual rate of $6\%.$ However, after $4$ years, the borrower needs an additional RM$2000$ and must restructure all outstanding debts over the remaining $6$ years at $7\%$ effective. Calculate the payment amount during those $6$ years.

$($b$)$ Using the amortization method, a person borrows RM$5,000$ at an effective rate of $8\%$ per annum and agrees to repay the loan with payments at the end of each year. The first payment is RM$600$ and each subsequent payment is $4\%$ above the previous one, with a smaller payment at the end of the term.

$($i$)$ Calculate the outstanding loan balance at the end of $5$ years.

$($ii$)$ Calculate the principal and the interest paid in the $5$th payment.