<|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
 The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least  $\sqrt{2}$ . It is known that a regular triangle with side lenght  $3$  cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of? 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
The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least  $\sqrt{2}$ . It is known that a regular triangle with side lenght  $3$  cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?
test