The mortgage on your house is $5$ years old. It required monthly payments of $$$$1423\mathrm{.}04,$ had an original term of $30$years$,$ and had an interest rate of $8\%($APR$).$ In the intervening $5$years$,$ interest rates have fallen and so you have decided to refinancelong dashthat $$is$,$ you will roll over the outstanding balance into a new mortgage. The new mortgage has a $30-$year $$term$,$ requires monthly$$ payments, and has an interest rate of $6\mathrm{.}125\%($APR$).$

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

b$.$ If you still want to pay off the mortgage in $25$years$,$ what monthly payment should you make after you$$ refinance?

c$.$ Suppose you are willing to continue making monthly payments of $$$$1423\mathrm{.}04.$ How long will it take you to pay off the mortgage after$$ refinancing?

d$.$ Suppose you are willing to continue making monthly payments of $$$$1423\mathrm{.}04,$ and want to pay off the mortgage in $25$ years. How much additional cash can you borrow today as part of the$$ refinancing?

$($Note$$: Be careful not to round any intermediate steps less than six decimal$$ places.$)$