
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
Daniel chooses a positive integer  $n$  and tells Ana. With this information, Ana chooses a positive integer  $k$  and tells Daniel. Daniel draws  $n$  circles on a piece of paper and chooses  $k$  different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the  $k$  points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of  $k$  that allows Ana to achieve her goal regardless of how Daniel chose the  $n$  circumferences and the  $k$  points.

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