
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
Given the volume of the container is $\frac{28}{3} \pi$ cubic meters, the height is 1 meter, and the construction costs per unit area for the cylindrical and hemispherical parts are 30,000 yuan and 40,000 yuan, respectively, find the radius $r$ when the construction cost of the container is minimized.

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