
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
Consider a circle with center \(O\) and a given radius. There are two tangents to the circle; one is at a distance \(r\) (radius of the circle) above the center, and the other is at distance \(2r\) below the center. Determine the number of points which are equidistant from the circle and both tangents.

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