Non-Splittable Subsets

Let \(S\) be a set of three integers chosen from \(N = \{1,2,\ldots,2012\}\). We say that \(S\) is 4-splittable if there is an \(n \in N \setminus S\) such that \(S \cup \{n\}\) can be partitioned into 2 sets such that the sum of each set is the same. How many 3-element subsets of \(N\) are not 4-splittable?

Details and assumptions

You can refer to Set Notation for the definition of \( N \setminus S\).

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