<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Find the largest positive integer $k$ such that the set of positive integers can be divided into $k$ subsets $A_{1}, A_{2}, \cdots, A_{k}$, so that for all integers $n(n \geq 15)$ and for all $i \in \{1, 2, \cdots, k\}$, there exist two distinct elements in $A_{i}$ whose sum is $n$. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
3
Find the largest positive integer $k$ such that the set of positive integers can be divided into $k$ subsets $A_{1}, A_{2}, \cdots, A_{k}$, so that for all integers $n(n \geq 15)$ and for all $i \in \{1, 2, \cdots, k\}$, there exist two distinct elements in $A_{i}$ whose sum is $n$.
test