$($a$)$ A company purchased a patent on January $1,2024,$ for $2,500,000,000.$ The patent's legal life is $20$ years but the company estimates that the patent's useful life will only be $5$ years from the date of acquisition. On June $30,2024,$ the company paid legal costs of $135,000,000$ in successfully defending the patent in an infringement suit. Prepare the journal entry to amortize the patent at year$-$end on December $31,2024.$

$($b$)$ Clark Company purchased a franchise from Tastee Food Company for $$400,000$ on January $1,2024.$ The franchise is for an indefinite period and gives Clark the exclusive rights to sell Tastee Wings in a particular territory. Prepare the journal entry to record the acquisition of the franchise and any necessary adjusting entry at year$-$end on December $31,2024.$

$($c$)$ Hulse Company incurred research costs of $$500,000$ in $2024$ in developing a new product. Prepare the necessary journal entries during $2024$ to record these events and any adjustments at year$-$end on December $31,2024.$