The interest rate for the first five years of Terilynn and Ross's $$120,000$ mortgage is $4\%$ compounded semi$-$annually. Monthly payments are based on a $25-$year amortization. After $2$ years of the term, Terilynn and Ross decided to take advantage of the privilege of increasing the payments on their mortgage by $10\%.$

a$)$ How much will the amortization period be shortened?

b$)$ What will be the principal balance at the end of the five$-$year term?