
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 1 inch long grows according to the following law, where the first term is the initial length.
\[1 + \frac{1}{3}\sqrt{3} + \frac{1}{3} + \frac{1}{9}\sqrt{3} + \frac{1}{9} + \frac{1}{27}\sqrt{3} + \frac{1}{27} + \cdots\]
Determine the limit of the length of the line if this growth process continues indefinitely.
A) $\frac{3 + 3 \sqrt{3}}{2}$
B) $\frac{3 + \sqrt{3}}{2}$
C) $\frac{9 + 2\sqrt{3}}{5}$
D) $\frac{3}{2} + \sqrt{3}$
E) $\frac{4 + \sqrt{3}}{2}$

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