A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $4$-player teams, but no teams may have exactly the same $4$ members. The site statistics show a curious fact: The product of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people, and the average, over all subsets of size $7$ of the set of $n$ participants, of the number of complete teams whose members are among those $7$ people, is equal to $2$. How many values $n$, $8\le n\le 2000$, can be the number of participants?