I've heard of a suggestion of using 13 months of 4 weeks each.

It adds up to 364 days.

The remainder day is a new type of annual leap day and you get the additional normal one every 4 years.

those come from the conflict between the solar and lunar calendars

we eat captured solar energy and rely on tides & animal behavior which is connected to lunar cycles, the sun and the moon are what our efforts to measure time are hinged on at the macro level. the biggest problem is that the sun and moon do not sync up neatly in their movements. there are solar calendars and lunar calendars and attempts to merge them are difficult mathematically.

for example:

Since each lunation is approximately 29+1⁄2 days, it is common for the months of a lunar calendar to alternate between 29 and 30 days. Since the period of 12 such lunations, a lunar year, is 354 days, 8 hours, 48 minutes, 34 seconds (354.36707 days), purely lunar calendars are 11 to 12 days shorter than the solar year. In purely lunar calendars, which do not make use of intercalation, the lunar months cycle through all the seasons of a solar year over the course of a 33–34 lunar-year cycle.