
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
A sphere with radius \( R \) and a right circular cone with a base radius \( R \) and height \( 2R \) rest in the same plane. These solids are intersected by a plane parallel to the base plane. At what distance from the base plane should the intersecting plane be such that:

1) The areas of the circles cut from the two solids are equal;
2) The volume of the truncated cone between the base and the intersecting plane is \( n \) times the volume of the spherical segment between the same planes.

For what values of \( n \) is the task possible?

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