
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
Given an infinite geometric sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n$. The first term of the sequence is the coefficient of $x$ in the expansion of the binomial $$(x+ \frac {1}{x})^{7}$$, and the common ratio is the modulus of the complex number $$z= \frac {1}{1+ \sqrt {3}i}$$, where $i$ is the imaginary unit. Find $$\lim_{n\rightarrow \infty} S_{n}$$.

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