Joseph planned to buy a house but could afford to pay only $$7,500$ at the end of every $6$ months for a mortgage with an interest rate of $4\mathrm{.}20\%$ compounded semi$-$annually for $25$ years. He paid $$28,750$ as a down payment.

a$.$ What was the maximum amount he could afford to pay for a house?

b$.$ What was his total amount spent for the house through the mortgage period including the downpayment $($not taking the time$-$value of money into account$)?$