[MOSP 2001] Counting number of sets

Let \( a_n \) denote the number of nonempty sets \( S \) such that:

(i)\( S \subseteq \{1,2,...,n\}\)

(ii) All elements of \( S\) have the same parity;

(iii) Each element \( k\in S\) such that \( k\geq 2|S|\).

What is the value of \( a_{37} \)?

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