The problem describes a debt to be amortized. $($Round your answers to the nearest cent.$)$

A man buys a house for $$360,000.$ He makes a $$150,000$ down payment and amortizes the rest of the purchase price with semiannual payments over the next $12$ years. The interest rate on the debt is $9\%,$ compounded semiannually.

$($a$)$ Find the size of each payment.

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$($b$)$ Find the total amount paid for the purchase.

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$($c$)$ Find the total interest paid over the life of the loan.