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.

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^3 $ ). Description of the test cases follows. Each of the next $ t $ lines describes one test case and contains two integers $ a $ and $ b $ ( $ 0 \le a, b \le 10^5 $ ; $ a + b > 0 $ ) — the number of games won by Alice and Borys, respectively. It is guaranteed that the sum of $ a + b $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case print two lines. In the first line, print a single integer $ m $ ( $ 1 \le m \le a + b + 1 $ ) — the number of values of $ k $ such that exactly $ k $ breaks could happen during the match. In the second line, print $ m $ distinct integers $ k_1, k_2, \ldots, k_m $ ( $ 0 \le k_1 < k_2 < \ldots < k_m \le a + b $ ) — the sought values of $ k $ in increasing order.

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.