When Julius Caesar reformed the calendar, why didn't he just create months with 30 or 31 days, resulting in 7*30+5*31=365 in regular years and 6*30*6*31=366 in leap years? Why he wanted February to be an exception?

I had a similar question. For some reason I had it in my mind that the length of the month had to do with the rotation of the moon around the Earth. It takes 27.3 days for the moon to circle the Earth. I always wondered why the months were not 28 days long. 28 days x 13 months = 364 days. They could have added a day on the end. Was it because 13 was considered an unlucky number?