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Look at the Jewish calendar. We solved this problem approx. 1670 years ago. You add a 14th month every 2 or 3 years.
https://www.jewfaq.org/jewish_calendar
The East Asian Lunar calendar also uses a similar system.
The reason the Greek-Roman-Christian calendar gave up on synching the months is that without a zero and some advanced algebra you can't calculate how to automate the calendar. That meant the Greeks and later Romans would have had to keep changing the calendar manually; instead they just dropped the lunar bit entirely and kept messing with it until the 1800s when they finally got the leap days right for it. The Roman-Christian calendar is kinda uniquely bad amongst cultural calendars and it was also uniquely slow to get it's drift compensation automated & correct.
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