Maximum Subsequence Sum

Chef has a machine which he uses to rotate sequences. If he puts a sequence $S_1, S_2, \ldots, S_N$ into the machine, it produces the sequence $S_2, S_3, \ldots, S_N, S_1$, i.e. the first element of the sequence is moved to the end. Chef is definitely not a newbie ― he knows about trivial things like finding the maximum sum of a contiguous subsequence. Therefore, he now made a difficult problem