Equal Pairs (Hard)

**This is the hard version of the problem. In this version, the array $A$ is changing. The constraint on $N$ is also larger: $N \le 2 \cdot 10^5$.** For an array $A$ containing non-negative integers, we define $f(A)$ as follows: - First, for each index $i$ such that $A_i = 0$, replace $A_i$ by an arbitrary **positive** integer of your choice. - Then, calculate the number of pairs $(i, j)$ such