There are \(2026\) people at a big party. Some pairs of people are friends (nobody is friends with themselves, and friendship goes both ways). Starting one minute after midnight, the host begins sending people home in a strange way.
After 1 minute, anyone with no friends at the party leaves. After 2 minutes, anyone with exactly one friend currently in the party leaves. After 3 minutes, anyone with exactly two friends currently in the party leaves, and so on.
(Important: at each minute, everyone checks their number of friends at the same time, and then all the people who qualify leave together.)
Show that after \(2026\) minutes have passed, at least two people will have left the party.