A particular commodity has a price$-$supply equation given by $p=369(1\mathrm{.}035{)}^x,$ where $x$ is the numbers of items of the commodity demanded when the price is $p$ dollars per item.

$($a$)$ Find producers' surplus if the equilibrium quantity is $53$ items. $($Round your answer to the nearest cent if necessary.$)$ $

$($b$)$ Find producers' surplus if the equilibrium price is $2,417$ dollars. $($Round your answer to the nearest cent if necessary.$)$ $