Solve the problem $(8$ points$)$:

Suppose a factory produces four different products, the first product has a daily output of $1,$ the second product has $2,$ and so on$.$ The goal is to determine the maximum daily production profit for each product, given the following conditions:

Profit per unit of product is $$20,$$$12,$$$40,$ and $$25$ for each of the four products, respectively.

Due to the lack of manpower, the total number of units produced per day cannot exceed $50.$

Each unit of product $1$ consumes $3$ units of raw material A$.$ Each unit of product $2$ requires $2$ units of raw material A and $1$ unit of raw material B$.$ Each unit of product $3$ requires $1$ unit of A and $2$ units of B$.$ Finally, each unit of $4$ th product requires three units. $b.$

Due to transportation and storage restrictions, the factory can consume up to $100$ units of raw material A and $90$ units of $B$ per day.