Adding on Segments

There are $N$ integers $A_1, A_2, \ldots, A_N$ on a circle. For each valid $i$, $A_i$ and $A_{i+1}$ are adjacent; $A_1$ and $A_N$ are also adjacent. Chef has a sequence of integers $B_1, B_2, \ldots, B_N$, which he may use to perform any number of operations (including zero). In one operation, he should do the following: - Choose an integer $i$ ($1 \le i \le N$). - Choose $i$ consecutive inte