
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

We inscribe a sphere around a cone. At what distance from the base of the cone should a plane parallel to the base be placed so that the area of the circle cut out of the sphere is twice the area of the circle cut out of the cone? The radius of the base of the cone is $r$, and the height of the cone is $m$.

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