
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]


system
Given is an integer  $k\geq 2$ . Determine the smallest positive integer  $n$ , such that, among any  $n$  points in the plane, there exist  $k$  points among them, such that all distances between them are less than or equal to  $2$ , or all distances are strictly greater than  $1$ .

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