$\textbf{Birthday}$

John and Mary know that their boss Joe's birthday is one of the following $10$ dates: Mar $4$, Mar $5$, Mar $8$, Jun $4$, Jun $7$, Sep $1$, Sep $5$, Dec $1$, Dec $2$, and Dec $8$.

Joe tells John only the month of his birthday, and tells Mary only the day.

After that, John first says: “I do not know Joe’s birthday, Mary doesn’t know it either.”

After having heard what John said, Mary says "I did not know Joe's birthday, but I know it now."

After having heard what Mary said, John says "I know Joe's birthday now as well."

What is Joe's birthday?