CF1558A Charmed by the Game

Alice and Borys are playing tennis. A tennis match consists of games. In each game, one of the players is serving and the other one is receiving. Players serve in turns: after a game where Alice is serving follows a game where Borys is serving, and vice versa. Each game ends with a victory of one of the players. If a game is won by the serving player, it's said that this player holds serve. If a game is won by the receiving player, it's said that this player breaks serve. It is known that Alice won $ a $ games and Borys won $ b $ games during the match. It is unknown who served first and who won which games. Find all values of $ k $ such that exactly $ k $ breaks could happen during the match between Alice and Borys in total.

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In the first test case, any number of breaks between $ 0 $ and $ 3 $ could happen during the match: - Alice holds serve, Borys holds serve, Alice holds serve: $ 0 $ breaks; - Borys holds serve, Alice holds serve, Alice breaks serve: $ 1 $ break; - Borys breaks serve, Alice breaks serve, Alice holds serve: $ 2 $ breaks; - Alice breaks serve, Borys breaks serve, Alice breaks serve: $ 3 $ breaks. In the second test case, the players could either both hold serves ( $ 0 $ breaks) or both break serves ( $ 2 $ breaks). In the third test case, either $ 2 $ or $ 3 $ breaks could happen: - Borys holds serve, Borys breaks serve, Borys holds serve, Borys breaks serve, Borys holds serve: $ 2 $ breaks; - Borys breaks serve, Borys holds serve, Borys breaks serve, Borys holds serve, Borys breaks serve: $ 3 $ breaks.