Assuming that the value of a property in a Toronto suburb would double over $25$ years, Arash would purchase a house worth $$600,000$ by making a down$-$payment of $$30,000$ and obtaining a mortgage for the balance amount from a local bank at an interest rate of $4\%$ compounded semi$-$annually for $25$ years.

a$.$ If the interest rate is constant over the $25-$year period, calculate the month$-$end payments for the mortgage. What would be his total investment in the house over the term?