Month-Week Calendar

You may have heard of the ISO week calendar, or just "the ISO calendar". Using weeks beginning on Monday,
it defines a week-year, and sets the first week in a year to be the first one with a majority of its days in January.
However, it has a problem: consider the date 2025W34-3, on which I am writing this text.
What month is it in? You can't tell just by looking at it... but we can combine it with months! So we get the "month-week" calendar.

The rule for the month-week calendar is that the first week of a month is the first week that a majority of
that month, or, equivalently, the first week that has its Thursday in the month.