Author ~Hamed_Ghaffari, While reading Problem E in today's Codeforces Round 1124 (Div. 2), I noticed an interesting visual similarity to a problem concept I was working on recently. While Problem E focuses on a fixed range with a range maximum, my version involves arbitrary array rearrangement and full partitioning into disjoint subsets.

You are given an array $$$A$$$ of $$$N$$$ integers. You may rearrange the elements of $$$A$$$ in any arbitrary order to form a new array $$$A'$$$.

After rearranging, you must divide $$$A'$$$ into $$$m$$$ ($$$m \ge 1$$$) contiguous, disjoint subarrays $$$S_1, S_2, \dots, S_m$$$ such that every element of $$$A'$$$ belongs to exactly one subarray.

Let $$$F(S)$$$ denote the bitwise AND of all elements in a subarray $$$S$$$:

Version 1 (Feasibility Check): Given an integer $$$K$$$, determine if there exists a valid rearrangement and partition such that:

Version 2 (Maximization): Find the maximum possible value of:

among all valid rearrangements and partitions of $$$A$$$ .