
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 all points $(x, y)$ in the coordinate plane that satisfy both $x^2 + (y - 5)^2 \leq 25$ and $y \ge x - 3$. Determine the volume of the solid formed when $\mathcal{S}$ is revolved around the line $y = x - 3$.

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