ILL 1983 Problem 36

The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that:(i) the union of any three of these subsets is the entire set $X$ , while**(ii)** the union of any two of them contains at most $1979$ members. What is the largest possible value of $k ?$