Range Maximum Deconstruction (Hard)

**This is the hard version of the problem. The only difference between this version and the easy version is that in this version, there is no constraint on the parity of $N$. This problem is worth $70$ points.** For an array $A$ of length $N$, its *range maximum* array, denoted $B$, is another array of length $N$ constructed as follows. For each index $i$ ($1 \leq i \leq N$), - Let $k = \min(i