
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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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?

Present the answer in LaTex format: \boxed{Your answer}
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