
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
Construct a cylindrical iron barrel with a volume of $V$. The lid of the barrel is made of aluminum alloy, and the price of aluminum alloy per unit area is three times that of iron. To minimize the cost of this container, the ratio of the bottom radius $r$ of the iron barrel to its height $h$ should be _______.

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