Catherine took out a $3-$year loan of $$24,700$ at a nominal rate of $6\%,$ compounded semiannually. Level interest payments are paid at the end of every half year, and the principal is repaid at the end of $3$ years by the accumulation of capital in a sinking fund that earns $5\%,$ compounded semiannually.

Catherine makes non$-$level sinking fund deposits. During the first $2$ years, the $4$ sinking fund deposits are $$4,500$ each. The remaining two deposits are of size $F.$

$($a$)$ Find $F.$

$($b$)$ Construct an amortization schedule that traces the progress of the sinking fund balance, the net loan balance, and also the net interest for each half year.