
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 line initially 2 inches long grows according to the following law, where the first term is the initial length.
\[2 + \frac{1}{2} + \frac{1}{3}\sqrt{3} + \frac{1}{4} + \frac{1}{9}\sqrt{3} + \frac{1}{16} + \frac{1}{27}\sqrt{3} + \frac{1}{64} + \cdots\]
If the growth process continues forever, find the limit of the length of the line.
A) $3 + \frac{1}{2}\sqrt{3}$
B) $3 + \frac{1}{3}\sqrt{3}$
C) $3 + \sqrt{3}$
D) $3 + \frac{3}{2}\sqrt{3}$
E) $4 + \frac{1}{2}\sqrt{3}$

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