$1.$ A company produces four different metal products. Each product goes through three separate processes: machining, grinding and joining. In the table below, the processing times required for each product are given in hours.

Product I

Machining: $3$

Grinding:$1$

Joining:$2$

Product II

Machining: $2$

Grinding: $1$

Joining: $2$

Product III

Machining: $2$

Grinding $2$

Joining:$2$

Product IV

Machining: $4$

Grinding: $3$

Joining: $1$

The company can perform $480$ hours of machining, $400$ hours of grinding and $400$ hours of joining per week. The selling prices of the products are $48,32,45$ and $64$ TL respectively. According to the agreement with the distributor, the firm must produce at least $50$ units of product I and at least $100$ units of product II or product III. It can also sell a maximum of $25$ units of product IV$.$

Construct a linear programming model to maximize the firm's total profit.