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A home gym is listed for $$812\mathrm{.}45$ less $35\%,12\%,5\%.$

$($a$)$ What is the net price? $($Round your answer to the nearest cent, if necessary. Do not include the dollar sign or commas in your answer.$)$

$($b$)$ What is the total amount of discount allowed? $($Round your answer to the nearest cent, if necessary. Do not include the dollar sign or commas in your answer.$)$

$($c$)$ What is the exact single rate of discount that was allowed? $($Round your answer to $2$ decimal places, if necessary. Do not include the percentage sign your answer.$)$