Statistics · Superchallenge problem

Seven whole numbers are written in order from smallest to largest. Their box plot shows: minimum $2$, first quartile $4$, median $7$, third quartile $10$, maximum $15$. (The first quartile is the median of the three numbers below the median, and the third quartile is the median of the three numbers above it.) What is the largest possible mean of the seven numbers? Give your answer as a fraction.