
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 place a sphere inside a hollowed equilateral cone such that the sphere touches the cone's surface at the circumference of the base. What fraction of the sphere's surface lies inside the cone and what fraction lies outside it?

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