Holiday Contest 2025 - Guts Round - Set 1 Problem 1

The Cotsworth calendar (also called the "International Fixed Calendar") is a calendar system in which every month has 28 days except for 1 month which has 29 days (there are 13 months total). However, in the world of Sbrtugimmurstyj, December 31st does not exist (Sbrtugimmurstyjies hate the New Year) so January 1st is always a Sunday (this means there are 13 months of 28 days each). In this world, there exists some number of people. What is the minimum number of people needed to exist such that in any month of a year, there exists one week containing at least three people's birthdays?