Alexa will amortize a $$35,000$ car loan through equal payments at the end of each month, for $54$ months. The interest rate is $8\mathrm{.}25\%$ compounded monthly, find:

$($a$)$ The amount of each payment.

$($b$)$ The outstanding principal at the beginning of the ${30}^{th}$ period.

$($c$)$ The interest in the ${30}^{th}$ payment.

$($d$)$ The principal in the ${30}^{th}$ payment.

$($e$)$ The total interest paid $($i$.$e$.$ the finance charge$).$