Statement

Edoardo and Luca always try to come up with fun games that involve mathematics to entertain themselves on the Saturday nights in this pandemic. They both have N empty slots to fill in with numbers. Before starting to play, the two friends agree on a special value K. The final goal of the game is to obtain two sequences with the same sum (modulo 10^9 + 7). The game is not easily over, as there are two very strict requirements: (1) Numbers cannot be repeated within the same sequence and cannot be present in both sequences; (2) The XOR (exclusive or) of the numbers in each sequence must be exactly K. Finding those sequences, however, comes out to be harder than expected. Can you help them?