The Helton Fine Art Gallery is considering installing a video camera security system to reduce its insurance premiums. A diagram of Helton's eight exhibition rooms is shown in the figure below; the openings between the rooms are numbered $1$ to $13.$ A security firm proposed that two$-$way cameras be installed at some room openings. Each camera has the ability to monitor the two rooms between which the camera is located. For example, if a camera were located at opening number $4,$ rooms $1$ and $4$ would be covered; if a camera were located at opening $12,$ rooms $2$ and $6$ would be covered; and so on$.$ Management decided not to locate a camera system at the entrance to the display rooms. The objective is to provide security coverage for all eight rooms using the minimum number of two$-$way cameras.

Formulate a binary integer linear programming model that will enable Helton's management to determine the locations for the camera systems. Solve the model to determine how many two$-$way cameras to purchase and where they should be located.