Jan invests in a fund by making a $$8,000$ deposit at the beginning of each year for $20$ years.

The fund grew at an annual effective interest rate of $9\%.$ At the end of $20$ years, she begins to withdraw

$$20,000$ from the fund semiannually, at the beginning of each six months. The amount of the final

withdrawal, will be the remaining balance of the fund, smaller than $$20,000,$ to deplete the fund.

$($a$)$ How many $$20,000$ withdrawals can Kim make?

$($b$)$ How much is the final withdrawal?