A company produces two types of products: Product A and Product B$.$ To

maximize profit, the company needs to determine the optimal quantities

of each product to produce. Product A costs $$100$ to manufacture per

unit, while Product B costs $$150$ per unit.

The company has limited resources: $400$ labor hours and $450$ units of raw

materials available for production. The production process requires $2$

labor hours for each unit of Product A and $3$ labor hours for each unit of

Product B$.$ Additionally, producing one unit of Product A requires $4$ units

of raw material, while producing one unit of Product B requires $3$ units of

raw material.

Find the solution to minimize the total production cost.