
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
Consider the infinite series $1 + \frac{1}{2} - \frac{3}{4} - \frac{3}{8} + \frac{9}{16} + \frac{9}{32} - \frac{27}{64} - \frac{27}{128} + \cdots$. Let $S$ be the (limiting) sum. Determine the value of $S$.
A) $\frac{1}{2}$
B) $\frac{4}{7}$
C) $\frac{6}{7}$
D) $\frac{3}{4}$
E) $\frac{8}{9}$

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