The mortgage on your house is $55$ years old. It required monthly payments of $$$$1$ comma $4021,402,$ had an original term of $3030$years$,$ and had an interest rate of $10\%10\%($APR$).$ In the intervening $55$years$,$ interest rates have fallen and so you have decided to refinancelong dash$$that $$is$,$ you will roll over the outstanding balance into a new mortgage. The new mortgage has a $3030-$year $$term$,$ requires monthly$$ payments, and has an interest rate of $6\mathrm{.}625\%6\mathrm{.}625\%($APR$).$

a$.$ What monthly repayments will be required with the new$$ loan?

b$.$ If you still want to pay off the mortgage in $2525$years$,$ what monthly payment should you make after you$$ refinance?

c$.$ Suppose you are willing to continue making monthly payments of $$$$1$ comma $4021,402.$ How long will it take you to pay off the mortgage after$$ refinancing?

d$.$ Suppose you are willing to continue making monthly payments of $$$$1$ comma $4021,402$ and want to pay off the mortgage in $2525$ years. How much additional cash can you borrow today as part of the$$ refinancing?