Suppose that you make a charge of $$1,000$ to a new credit card, but then make no other charges. Assume that the finance charges amount to a monthly rate of r as a decimal. Let the minimum payment as a percentage of the new balance be given by the number m when expressed as a decimal. Then your balance B$,$ in dollars, is an exponential function of the number of payments n given by the formula B $=1,000$ an$,$ with initial value $$1,000$ and base a $=(1+$ r$)(1$ m$).$

$($a$)$ Find the exact value of the base a if the monthly finance charge is $1\mathrm{.}3\%$ and the minimum payment is $4\%$ of the new balance.

a $=$

$($b$)$ In the situation of part $($a$),$ by what percentage does your balance decrease each month? $($Round your answer to two decimal places.$)$

$($c$)$ What is the exact value of the base a if the monthly finance charge is $1\mathrm{.}5\%$ and the minimum payment is $0\mathrm{.}9\%$ of the new balance? $($Round your answer to five decimal places.$)$

Is thias a realistic situation?