Katherine received a $15$ year loan of $$335,000$ to purchase a house. The interest rate on the loan was $2\mathrm{.}10\%$ compounded semi$-$annually.

a$.$ What the size of the monthly loan payment?

Round to the nearest cent

b$.$ What is the balance of the loan at the end of year $4?$

Round to the nearest cent

c$.$ By how much will the amortization period shorten if Katherine makes an extra payment of $$30,000$ at the end of year $4?$