Question $3.$ A basketball coach needs to choose five players to form the starting lineup of his team. The team consists of seven players in total, who have been rated on a scale of $1($poor$)$ to $3($excellent$)$ according to their ball handling, shooting, rebounding, and defensive abilities. There are three positions: Guard $($G$),$ Center $($C$),$ and Forward $($F$).$ The positions that each player is capable of playing and the players abilities are summarized in the following table:

The five$-$player starting lineup must satisfy the following restrictions:

$1.$ At least $4$ players must be able to play guard, at least two players must be able to play forward, and at least $1$ player must be able to play center.

$2.$ The average ball handling, shooting, and rebounding level of the starting lineup must be at least $2.$

$3.$ If player $3$ is included, player $6$ cannot be included.

$4.$ If player $1$ is included, then players $4$ and $5$ must both be included.

$5.$ Either player $2$ or player $3$ must be included.

Given these constraints, formulate an integer program that maximizes the defensive ability of the starting lineup $($i$.$e$.$ maximizes total defensive points$).$