Problem $3(20$ points$).$ A company assembles two products: A and B$.$ Product A sells for $\backslash (\backslash$$ $11\backslash )$ per unit, and product B sells for $\backslash (\backslash$$ $23\backslash )$ per unit. A unit of product A requires $2$ hours on assembly line $1$ and $1$ unit of raw material. A unit of product B requires $2$ units of raw material, $1$ unit of A$,$ and $2$ hours on line $2.$ For line $1,1,300$ hours of time are available and $500$ hours of time are available on line $2.$ A unit of raw material may be bought $($for $\backslash (\backslash$$ $5\backslash )$ a unit$)$ or produced $($at no cost$)$ by using $2$ hours of time on line $1.$ Formulate a LO model to maximize the profit for the company, and solve the problem using your favorite method $($either through computer tools or directly using the simplex method$).$