Transmitting Levels

Optimizing the amount of data transmitted via a network is an important and interesting part of developing any network application.

In one secret game developed deep in the ZeptoLab company, the game universe consists of n levels, located in a circle. You can get from level i to levels i - 1 and i + 1, also you can get from level 1 to level n and vice versa. The map of the i-th level description size is a_i bytes.

In order to reduce the transmitted traffic, the game gets levels as follows. All the levels on the server are divided into m groups and each time a player finds himself on one of the levels of a certain group for the first time, the server sends all levels of the group to the game client as a single packet. Thus, when a player travels inside the levels of a single group, the application doesn't need any new information. Due to the technical limitations the packet can contain an arbitrary number of levels but their total size mustn't exceed b bytes, where b is some positive integer constant.

Usual situation is that players finish levels one by one, that's why a decision was made to split n levels into m groups so that each group was a continuous segment containing multiple neighboring levels (also, the group can have two adjacent levels, n and 1). Specifically, if the descriptions of all levels have the total weight of at most b bytes, then they can all be united into one group to be sent in a single packet.

Determine, what minimum number of groups do you need to make in order to organize the levels of the game observing the conditions above?

As developing a game is a long process and technology never stagnates, it is yet impossible to predict exactly what value will take constant value b limiting the packet size when the game is out. That's why the developers ask you to find the answer for multiple values of b.