Katie has a $$40,000$ student loan at $3\mathrm{.}65\%$ compounded semi$-$annually amortized over $9$ years with payments made at the end of every month.

a$.$ What is the size of Katie's payments at the end of every month?

b$.$ What was the principal portion of payment $36?$

c$.$ What was the size of Katie's' final payment?