Volunteers are preparing balloons for the 10th anniversary of SUSTech. The volunteers found that the balloons they buy are all strung into circles (see the figure below). The balloons are connected by ropes, and the volunteer need to use scissor to cut some ropes to separate the circle into parts. For example, for a circle with 4 balloons, at least 2 ropes need to be cut if we want to get 2 separated parts, and each part has 2 balloons. Now, the volunteers find that there are \(n\) balloon circles in the warehouse, the \(i\)-th circle has \(c_i\) balloons, and there are \(m\) different rooms that need decoration. So what is the least number of ropes the need to be cut so that every room can be decorated by at least two rope-connected balloons?