We call $A_1, A_2, \ldots, A_n$ an $n$-division of $A$ if:

1. $A_1 \cap A_2 \cap \cdots \cap A_n = A$,
2. $A_i \cap A_j \neq \emptyset$ for $i \neq j$.

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 $A_i$ where $b < a \leq \frac{4}{3}b$.

Let's think step by step and output the final answer within \boxed{}.
user