
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
Let $\mathcal{S}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|9 - x| + y \le 12$ and $3y - x \ge 18$. When $\mathcal{S}$ is revolved around the line whose equation is $3y - x = 18$, the volume of the resulting solid is $\frac{m\pi}{n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p$.

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