
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 in the coordinate plane defined by the inequalities $|6 - x| + y \le 8$ and $4y - x \ge 20$. When $\mathcal{S}$ is revolved around the line $y = x/4 + 5$, compute the volume of the resulting solid in terms of $\pi$. Assume the region forms a triangle.

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