Minimize OR of Remaining Elements Using Operations

Given an array where you can perform up to k adjacent merges that replace two neighbors with their bitwise AND, find the minimum possible bitwise OR of the final elements. With n up to 1e5 and 30-bit values, the core challenge is choosing which adjacent segments to merge (interval/grouping decisions) so as to clear as many bits as possible (often handled via per-bit feasibility checks or DP).