A $$111,500$ mortgage for $20$ years for a new home is obtained at the rate of $6\mathrm{.}4\%$ compounded monthly. Find $($a$)$ the monthly payment, $($b$)$ the interest in the first payment, $($c$)$ the principal repaid in the first payment, and $($d$)$ the finance charge.

$($a$)$ The monthly payment on the mortgage is $ $$

$($Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places as needed.$)$