Automotive Excellence Inc. borrowed $$16,000\mathrm{.}00$ on May $11$ with an interest rate of $4\mathrm{.}5\%$ per annum. On July $21,$ $$4700\mathrm{.}00$ was repaid, and on August $12,$$$5200\mathrm{.}00$ was repaid. Automotive Excellence paid the balance of the loan on October $6.$ What was the final payment?

The final payment was $

$($Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.$)$