A borrower of $$25,000$ agrees to pay back the loan with $16$ level annual payments with the first payment due at the time the loan is issued. The annual effective rate of interest is $3\%$ for the first $7$ years and $5\%$ for the remaining years.

After the $8$th payment, the loan is refinanced. The annual payments are increased by $$500$ paid in arrears, and the number of annual payments is decreased accordingly. The final payment is larger than these new increased payments. Determine the last balloon payment.