<|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
 Among any five vertices of a standard grid, there are always two vertices such that the midpoint of the segment connecting them is also a vertex of the grid. What is the minimum number of vertices from a hexagonal grid that must be taken so that among them there are always two vertices such that the midpoint of the segment connecting them is also a vertex of this grid? 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
9
Among any five vertices of a standard grid, there are always two vertices such that the midpoint of the segment connecting them is also a vertex of the grid. What is the minimum number of vertices from a hexagonal grid that must be taken so that among them there are always two vertices such that the midpoint of the segment connecting them is also a vertex of this grid?
test