$(25$ points$)$ At a local business school, a toasted sandwich process uses a conveyor$-$fed oven. Zeynep is the sole operator of the sandwich$-$making process. In the first step of the process, she spends $2$ minutes putting various ingredients in the sandwich. Then she puts the sandwich on a conveyor belt and, over a period of $12$ minutes, the conveyor moves the sandwich from the beginning of the oven to the end of the oven, fully toasting it$.$ After the sandwich comes out of the oven, Zeynep spends $1$ minute slicing the sandwich and putting it in a box. At most, five sandwiches can fit in the oven at once. The toasting time in the oven does not depend on the number of sandwiches in the oven.