Margarita borrows $$14,000$ from her uncle and agrees to repay it in monthly installments of $$200.$ Her uncle charges $0\mathrm{.}6\%$ interest per month on the balance.

$($a$)$ If her balance $A_n$ in the $n$th month is given recursively by $A_0=14,000$ and $A_n=k*A_{n-1}-200,$ what is $k?$

$k=$

$($b$)$ Find her balance after seven months. $($Round your answer to the nearest cent.$)$