I believe theThe answer is yes. Consider the sequence
100, 200, 201, 202, 500, 601, 700, 701, 801, 1000, 1194, 1200.$100, 200, 201, 202, 500, 601, 700, 701, 801, 1000, 1194, 1200.$
It is easy to see that the set indexed by {1$X=\{1,2,5,7,10,12\}$ and $Y=\{3,4,6,8,9,11\}$ are exact. Moreover,2 we claim that they are the only exact subsets. To see this,5 note that for every subset $X'$ of $X$,7 the sum of the corresponding sequence values is divisible by $100$. However,10 for every proper subset $Y'$ of $Y$,12} the sum of the corresponding sequence values is not divisible by $100$. Thus, $X$ and $Y$ are the uniqueonly exact subset (up to complementation)subsets.