The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that:

1. The union of any three of these subsets is the entire set $X$.
2. The union of any two of them contains at most $1979$ members.

What is the largest possible value of $k$?

Let's think step by step and output the final answer within \boxed{}.
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