
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 pyramid with a square base and a height of 4 inches has its base side measuring 6 inches. The entire surface of the pyramid, including its base, is painted. A plane parallel to the base divides the pyramid into two solids, a smaller pyramid-shaped solid $P$ and a frustum-shaped solid $F$. This division is made such that the ratio of the areas of the painted surfaces of $P$ and $F$, as well as the ratio of the volumes of $P$ and $F$, are both equal to $k$. Find $k = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and determine $m+n$.

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