Kimiko signed a mortgage requiring payments of $$237\mathrm{.}49$ at the end of every month for $2$ years at $7\mathrm{.}8\%$ compounded monthly.

$($a$)$ How much was the original mortgage balance?

$($b$)$ If Kimiko missed the first $10$ payments, how much would she have to pay after $11$ months to bring the mortgage payments up to date?

$($c$)$ How much would Kimiko have to pay after $11$ months to pay off the mortgage $($assuming she missed all the payments$)?$

$($d$)$ If the mortgage were paid off after $11$ months, what would the total interest cost be$?$

$($e$)$ How much of the total interest cost is additional interest because of the missed payments?

$($a$)$ The original mortgage balance was $$.$

$($Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.$)$