
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 4 inches tall and its base has a 3-inch radius. 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  $k$ . Given that  $k=m/n$ , where  $m$  and  $n$  are relatively prime positive integers, find  $m+n$ .

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