
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 a new law, where the first term is the initial length and an additional linear term is introduced.
\[1+\frac{1}{3}\sqrt{3}+2\cdot\frac{1}{9}+3\cdot\frac{1}{27}\sqrt{3}+4\cdot\frac{1}{81}+\cdots\]
Determine the limit of the length of the line if the growth process continues indefinitely.
A) $\frac{1}{72}(4+\sqrt{3})$
B) $\frac{1}{3}(4+\sqrt{3})$
C) $\sqrt{3}$
D) $\infty$

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