CF526E 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 $ .

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In the test from the statement you can do in the following manner. - at $ b=7 $ you can divide into two segments: $ 2|421|32 $ (note that one of the segments contains the fifth, sixth and first levels); - at $ b=4 $ you can divide into four segments: $ 2|4|21|3|2 $ ; - at $ b=6 $ you can divide into three segments: $ 24|21|32| $ .