Maximum Number of Ways to Partition an Array

Given nums and a value k, find the maximum number of pivot indices (1 ≤ pivot < n) where the sum of elements left of pivot equals the sum right of pivot after you may change at most one element to k. The core challenge is to evaluate, using prefix sums and frequency counting, how replacing each element by k shifts left/right sums and thus creates or destroys valid pivots efficiently for n up to 1e5.