$5\mathrm{.}5$ A loan of amount $$3,000$ is to be repaid with bi$-$monthly payments of $$269\mathrm{.}36$ for $2$ years.

$($a$)$ Find the nominal rate of interest and the outstanding loan balance im$.$ mediately after the sixth payment is made.

$($b$)$ Explain why the answer in $($a$)$ is larger than one half of the initial loan