In one of the hotels on the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. The following information is known about the guests:

1. If two women know each other, then their husbands also know each other, and vice versa.
2. Each sauna can be used by either men only or women only (there is no limit on the number of people using a sauna at once, as long as they are of a single gender).
3. Each woman is only willing to share a sauna with women whom she knows.
4. Each man is only willing to share a sauna with men whom he does not know.

What is the greatest possible $k$ for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?

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