CF1956E2 Nene vs. Monsters (Hard Version)

This is the hard version of the problem. The only difference between the versions is the constraints on $ a_i $ . You can make hacks only if both versions of the problem are solved. Nene is fighting with $ n $ monsters, located in a circle. These monsters are numbered from $ 1 $ to $ n $ , and the $ i $ -th ( $ 1 \le i \le n $ ) monster's current energy level is $ a_i $ . Since the monsters are too strong, Nene decided to fight with them using the Attack Your Neighbour spell. When Nene uses this spell, the following actions happen in the following order one by one: - The $ 1 $ -st monster attacks the $ 2 $ -nd monster; - The $ 2 $ -nd monster attacks the $ 3 $ -rd monster; - $ \ldots $ - The $ (n-1) $ -th monster attacks the $ n $ -th monster; - The $ n $ -th monster attacks the $ 1 $ -st monster. When the monster with energy level $ x $ attacks the monster with the energy level $ y $ , the energy level of the defending monster becomes $ \max(0, y-x) $ (the energy level of the attacking monster remains equal to $ x $ ). Nene is going to use this spell $ 10^{100} $ times and deal with the monsters that will still have a non-zero energy level herself. She wants you to determine which monsters will have a non-zero energy level once she will use the described spell $ 10^{100} $ times.

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N/A

In the first test case, the following actions happen during the first $ 3 $ uses of the spell in this order: - Nene uses the Attack Your Neighbour spell for the first time; - the $ 1 $ -st monster attacks the $ 2 $ -nd monster, after the attack the energy level of the $ 2 $ -nd monster becomes equal to $ \max(0, 5-2)=3 $ ; - the $ 2 $ -nd monster attacks the $ 3 $ -rd monster, after the attack the energy level of the $ 3 $ -rd monster becomes equal to $ \max(0, 3-3)=0 $ ; - the $ 3 $ -rd monster attacks the $ 1 $ -st monster, after the attack the energy level of the $ 1 $ -st monster becomes equal to $ \max(0, 2-0)=2 $ ; - Nene uses the Attack Your Neighbour spell for the second time; - the $ 1 $ -st monster attacks the $ 2 $ -nd monster, after the attack the energy level of the $ 2 $ -nd monster becomes equal to $ \max(0, 3-2)=1 $ ; - the $ 2 $ -nd monster attacks the $ 3 $ -rd monster, after the attack the energy level of the $ 3 $ -rd monster becomes equal to $ \max(0, 0-1)=0 $ ; - the $ 3 $ -rd monster attacks the $ 1 $ -st monster, after the attack the energy level of the $ 1 $ -st monster becomes equal to $ \max(0, 2-0)=2 $ ; - Nene uses the Attack Your Neighbour spell for the third time; - the $ 1 $ -st monster attacks the $ 2 $ -nd monster, after the attack the energy level of the $ 2 $ -nd monster becomes equal to $ \max(0, 1-2)=0 $ ; - the $ 2 $ -nd monster attacks the $ 3 $ -rd monster, after the attack the energy level of the $ 3 $ -rd monster becomes equal to $ \max(0, 0-0)=0 $ ; - the $ 3 $ -rd monster attacks the $ 1 $ -st monster, after the attack the energy level of the $ 1 $ -st monster becomes equal to $ \max(0, 2-0)=2 $ . After each of the next uses of the spell, energy levels of monsters do not change. Thus, only the $ 1 $ -st monster has a non-zero energy level in the end. In the second test case, both monsters initially have zero energy level.