Circular Merging

$N$ integers $A_1, A_2, \ldots, A_N$ are placed in a circle in such a way that for each valid $i$, $A_i$ and $A_{i+1}$ are adjacent, and $A_1$ and $A_N$ are also adjacent. We want to repeat the following operation exactly $N-1$ times (until only one number remains): - Select two adjacent numbers. Let's denote them by $a$ and $b$. - Score $a+b$ penalty points. - Erase both $a$ and $b$ from the cir