
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 $\mathcal{P}$ be the region consisting of points in the coordinate plane that satisfy both $|6 - x| + y \le 8$ and $4y - x \ge 20$. When $\mathcal{P}$ is revolved around the line whose equation is $4y - x = 20$, find the volume of the resulting solid. Express the volume in the form $\frac {a\pi}{b\sqrt {c}}$, where $a$, $b$, and $c$ are positive integers, $a$ and $b$ are relatively prime, and $c$ is not divisible by the square of any prime.

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