
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
The area of a circle increases by a factor of 3 when its radius \( r \) is increased by \( n \). Determine the value of \( r \) in terms of \( n \).
A) \( \frac{n(\sqrt{3} + 1)}{2} \)  
B) \( \frac{n(\sqrt{3} - 1)}{2} \)  
C) \( n(2 - \sqrt{3}) \)  
D) \( \frac{n(3 - \sqrt{3})}{2} \)

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