On June $1,$ the unpaid balance in an account was $$177.$ On June $13,$ a payment of $$75$ was made. The finance charge rate was $1\mathrm{.}5\%$ per month of the average daily balance. Find the new balance at the end of June.

Group of answer choices

$$103\mathrm{.}98$

$$103\mathrm{.}93$

$$121\mathrm{.}80$

$$103\mathrm{.}50$