We call $ A_1, A_2, \ldots, A_n$ an $ n$-division of $ A$ if (i) $ A_1 \cap A_2 \cap \cdots \cap A_n = A$, (ii) $ A_i \cap A_j \neq \emptyset$. Find the smallest positive integer $ m$ such that for any $ 14$-division $ A_1, A_2, \ldots, A_{14}$ of $ A = \{1, 2, \ldots, m\}$, there exists a set $ A_i$ ($ 1 \leq i \leq 14$) such that there are two elements $ a, b$ of $ A_i$ such that $ b < a \leq \frac {4}{3}b$.