
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
If the sum of the first n terms of the sequence $\{a_n\}$ is $S_n$, and it satisfies the equation $a_n + 2S_n = 3$.  
(1) Can we find any three terms in the sequence that form an arithmetic sequence in their original order? Explain your reasoning.  
(2) Is it possible to extract an infinite geometric sequence from the sequence, such that the sum of all its terms, $S$, satisfies $\frac{9}{160} < S < \frac{1}{13}$? If such a sequence exists, how many of these geometric sequences are there?

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