John borrowed $$2,000$ on May ${23}^{rd}$ at an annual simple interest rate of $4,5\%.$ The length of the loan was $120$ days. On July ${13}^{th},$ he made a partial payment of $$800.$

a$)$ Determine the maturity date of the loan.

b$)$ Determine the amount of interest paid and the new balance due on July ${13}^{th}.$

c$)$ Determine the amount of the final payment, including interest, on the maturity date of the loan.