Find the smallest positive integer \(N\) with the following property. For any subset \(S\) of the set \(\{1,2,...,N\}\), there is an increasing three-term arithmetic progression of numbers from \(1\) to \(N\) such that either all of its terms or none of its terms are in \(S.\)

**Details and assumptions**

As an explicit example, if \(N = 1000 \) and \(S\) is the subset of all primes less than 1000, then the arithmetic progression \( 4-6-8 \) has none of its terms in \(S\).

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