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 \(k\)?