
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 solid in the shape of a right circular cone is 6 inches tall and its base has a radius of 5 inches. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid \( C \) and a frustum-shaped solid \( F \), in such a way that the ratio between the areas of the painted surfaces of \( C \) and \( F \) and the ratio between the volumes of \( C \) and \( F \) are both equal to \( k \). Find \( k \) if the height of the smaller cone \( C \) is 2 inches.

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