The Greens bought a $$257,000$ house. They made a down payment of $$48,000$ and took out a mortgage for the rest. Over the course of $30$ years they made monthly payments of $$1253\mathrm{.}07$ on their mortgage until it was paid off.

$($a$)$ What was the total amount they ended up paying for the house $($including the down payment and monthly payments$)?$

$$$

$($b$)$ How much interest did they pay on the mortgage?

$$$