$1(10\%)$ A company considers producing four new products. Managers need to determine the type

and amount of products to be manufactured in order to maximize total profit. The following

table shows the setup costs and marginal profit of each product.

Let $x_1,x_2,x_3,$ and $x_4$ represent the production rate of products $1,2,3,$ and $4,$ respectively. There

are three conditions that need to be satisfied.

$(1)$ No more than two of the products can be produced.

$(2)$ Either product $3$ or $4$ can be produced only if either product $1$ or $2$ is produced.

$(3)$ Either $5x_1+3x_2+6x_3+4x_{4}6000$ or $4x_1+6x_2+3x_3+5x_{4}6000$ is satisfied.

Introduce auxiliary binary variables to formulate a mixed BIP model for this problem.