Second MEX

We define the $\operatorname{SMEX}$ value of a sequence as the $2^{nd}$ smallest non-negative integer that does not appear in the array. For example, $\operatorname{SMEX}([0, 1, 2, 3]) = 5$ and $\operatorname{SMEX}([1, 1]) = 2$. You are given an array $A$ of $N$ elements. Calculate the sum of $\operatorname{SMEX}$ values over all $2^N - 1$ non-empty subsequences$^{\dagger}$ of $A$. Since the an