Every possible team (subset) of the players in ids, including the empty team, gets a badge number equal to the XOR of its members' ids (the empty team gets 0). Return the sum of all 2**n badge numbers modulo 1_000_000_007.
Examples
Input: ids = [1, 3]
Output: 6
Explanation: {} -> 0, {1} -> 1, {3} -> 3, {1, 3} -> 2. Sum 6.
Input: ids = [2, 2]
Output: 4
Constraints
0 <= len(ids) <= 10**5,0 <= ids[i] < 2**30.- Target complexity: O(n); enumerating subsets is impossible.
Goals
- Reason about each bit independently across all subsets
- Show that a bit present anywhere is set in exactly half of the subsets
- Reduce the answer modulo a prime with fast powering