A $$500,000$ house loan has been approved! Batrez Bank has given you the $30$ year$4\mathrm{.}00\%$ compounded monthly interest rate loan.

It's been exactly $57$ months after you bought the house. Your job has finally noticed all your hard work and dedication and has given you a big promotion and a big raise. You decide to pay an additional $$800$ per month starting this month to try to pay off quicker.

After exactly $60$ more monthly payments, you decide to refinance your loan. Morales Bank will add a charge of $1\%$ of the current balance and then lower your interest rate to $1\mathrm{.}90\%$ for the remaining loan. You will continue to pay the same amount as you did the month before.

What is the balance of the loan before you refinance?

What is the balance of the loan after you refinance?

How many months will it take to pay off the loan after you refinance?

Calculate the final payment of the home loan.