Tesla high school has 1000 students and \(k\) student clubs. You are told that:

- Each club has an odd number of members.
- The intersection of any two distinct clubs is an even (possibly zero) number of members.

For example, the pottery club might have 5 members, the science club might have 13 members, and there might be exactly 2 students who are in both clubs.

What is the maximum possible value of \(k\)?

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