
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 $xOy$, the shape enclosed by the line $y=x$, the line $x=1$, and the $x$-axis is rotated around the $x$-axis to form a cone. The volume of the cone $V_{\text{cone}}=\int_{0}^{1}{\pi {x}^{2}}dx=\frac{\pi }{3}\left. {{x}^{3}} \right|_{0}^{1}=\frac{\pi }{3}$. By analogy: the shape enclosed by the curve $y=2\ln x$, the line $y=1$, and the $x$ and $y$ axes is rotated around the $y$-axis to form a solid of revolution. The volume of this solid $V=$ \_\_\_\_\_\_\_\_.

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