Alyssa's Custom Cakes currently sells $5$ birthday, $1$ wedding, and $4$ specialty cakes each month for $$45,$$$155,$ and $$105$ each, respectively. Alyssa's current multifactor productivity ratio is $1\mathrm{.}25.$

$($a$)$ Use the multifactor productivity ratio provided to calculate the average cost of the cakes produced.

$($b$)$ Calculate Alyssa's labor productivity ratio in dollars per hour for each type of cake.

$($c$)$ Based solely on the labor productivity ratio, which cake should Alyssa try to sell the most.

$($d$)$ Based on your answer in part $($a$),$ is there a type of cake Alyssa should stop selling?