
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 manufacturer produces and sells a certain beverage in spherical bottles. The cost to manufacture a bottle is $0.8πr^{2}$ cents, where $r$ is the radius of the bottle in centimeters. It is known that for every $1mL$ of beverage sold, the manufacturer makes a profit of $0.2$ cents. The manufacturer can produce bottles with a maximum radius of $6cm$. What should the radius of the bottle be to minimize the profit per bottle?

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