Sepia Inc. issued bonds for $$475,000$ that were redeemable in $6$ years. They established a sinking fund that was earning $5\mathrm{.}65\%$ compounded semi$-$annually to pay back the principal of the bonds on maturity. Deposits were being made to the fund at the end of every $6$ months.

a$.$ Calculate the size of the periodic sinking fund deposit.

Round your answer up to the next cent

b$.$ Calculate the sinking fund balance at the end of the payment period $6.$

Round to the nearest cent

c$.$ Calculate the interest earned in payment period $7.$

Round to the nearest cent

d$.$ Calculate the amount by which the sinking fund increased in payment period $7.$