Let n be a positive integer, and let \mathcal{F} be a family of subsets of \left\{1,2, \cdots, 2^{n}\right\} such that for any non-empty A \in \mathcal{F} there exists B \in \mathcal{F} so that |A|=|B|+1 and B \subset A. Suppose that \mathcal{F} contains all \left(2^{n}-1\right)-element subsets of \left\{1,2, \cdots, 2^{n}\right\}. Determine the minimal possible value of |\mathcal{F}|.