
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 right circular cone is 6 inches tall with a base radius of 5 inches. The cone is fully painted and then cut by a plane parallel to the base, creating two parts: a smaller cone and a frustum. The ratios of the areas of their painted surfaces, as well as the volumes of these two parts, are both equal to $k$. If $k = \frac{m}{n}$ where $m$ and $n$ are coprime integers, find $m+n$.

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