
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
Design an open-top cylindrical barrel with a volume of $\frac {\pi}{2}$ cubic meters. The cost of the metal sheet for the base is 30 yuan per square meter, and for the side (lateral surface) is 20 yuan per square meter. To minimize the cost of materials, what should be the dimensions of the barrel's base radius and height?

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