Multiuser dungeon games, also called MUD games, are real-time virtual-world multiplayer games that are usually played online with text-based commands. The players do not meet normally, but every year, there is the Annual Congress of MUD (ACM) where MUD lovers all around the world could meet in person.
ACM is so popular that the event each year spans around 20 days. Each day, there will be a special gathering for MUD game designers to introduce their new games to the others.
Each player will usually spend a few days on the ACM site, and in-between will be invited in exactly one day to join this special gathering.
This year, ACM is held at your city, and your boss is an organiser, and he wants to find a best way to assign the players to these special gatherings (one player to one special gathering within his or her duration of stay), so that the maximum number of players among all gatherings is minimized.
Your boss is an extremely impatient guy. He wants to have a system that can tell the maximum number of players among all gatherings, in any best assignment, after each player enters his or her duration of stay. Your task is to help your boss develop such a system.