
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
A cylinder has a base radius of $\frac{\frac{3}{2}}{2}$ and a height of $\frac{{ }^{\frac{1}{2}+1}}{}$. Four identical spheres are inscribed in the cylinder such that they touch the upper base of the cylinder, its lateral surface, and each sphere touches two of the three other spheres. Find the lateral surface area of a cone whose base coincides with the lower base of the cylinder and which touches all four spheres.

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