
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
In the Cartesian coordinate system $xOy$, rotating the region bounded by the line $y=x$, the line $x=1$, and the $x$-axis around the $x$-axis for one full turn forms a cone. The volume $V_{\text{cone}}$ of the cone is $V_{\text{cone}}= \int_{0}^{1}\pi x^{2}dx= \frac{\pi}{3}x^{3}\bigg|_{0}^{1}= \frac{\pi}{3}$. By analogy, rotating the region bounded by the curve $y=2\ln x$, the line $y=1$, the $x$-axis, and the $y$-axis around the $y$-axis for one full turn forms a solid of revolution. The volume $V$ of this solid of revolution is $V=\boxed{\hspace{3cm}}$．

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