
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
Consider a shaded region on an $xy$-coordinate plane consisting of one rectangle of dimensions $2 \times 3$ units and one square of dimensions $2 \times 2$ units. The rectangle touches both the $x$-axis and $y$-axis, and the square is positioned such that one of its sides lies along the y-axis, starting immediately above the rectangle. If this entire region is rotated about the $y$-axis, what is the volume of the resulting solid in cubic units? Express your answer in simplest form in terms of $\pi$.

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