
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 bounded by the inequalities $|6 - x| + y \le 8$ and $2y - x \ge 10$. Find the volume of the solid formed when $\mathcal{S}$ is revolved around the line $2y - x = 10$. Express the volume as $\frac{a\pi}{b\sqrt{c}}$, where $a$, $b$, and $c$ are positive integers with $a$ and $b$ relatively prime and $c$ not divisible by the square of any prime. Determine $a + b + c$.

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