<image>Zhao and Wang are talking about their summer vacation activities. Zhao said, "My family went on a trip for a week, and the sum of the dates for those $$7$$ days is $$84$$. Can you figure out which day we left?" Wang said, "I stayed at my uncle's house for $$7$$ days during the summer vacation, and the sum of the calendar date and the month number is also $$84$$. Can you guess the date I returned home? Try to set up equations to solve Zhao's and Wang's problems. (Hint: Summer vacation is from July 1st to September 1st.)