A recent college graduate buys a new car by borrowing $$16,000$ at $8\mathrm{.}4\%,$ compounded monthly, for $5$ years. She decides to pay $$353$ instead of the monthly payment required by the loan.

$($a$)$ What is the monthly payment required by the loan? $($Round your answer to the nearest cent.$)$

How much extra did she pay per month? $($Round your answer to the nearest cent.$)$

$($b$)$ How many $$353$ payments will she make to pay off the loan? $($Round your answer up to the next whole number.$)$

$($c$)$ How much will she save by paying $$353$ per month rather than the required payment? $($Round your answer to the nearest cent.$)$