A mortgage for a condominium had a principal balance of $$41,200$ that had to be amortized over the remaining period of $6$ years. The interest rate was fixed at $4\mathrm{.}02\%$ compounded semi$-$annually and payments were made monthly.

a$.$ Calculate the size of the payments.

b$.$ If the monthly payments were set at $$745,$ calculate the size of the final payment.