Sortable Subarrays (Hard)

**This is the hard version of the problem. The only difference is the constraints. Here $N$ and sum of $N$ are bounded by $2 \cdot 10^5$.** An array $A$ is called *sortable* if it can be sorted in **strictly ascending order** with the use of the following operation any number of times: - Choose index $i$ and a **positive integer** $X$, and then update $A_i$ to $A_i \mod X$. Strictly ascending o