
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 that in the geometric sequence $\{a_n\}$, each term is positive, and the common ratio $q$ satisfies $0 < q < \frac{1}{2}$.

(1) Does there exist three terms in the sequence $\{a_n\}$ that form an arithmetic sequence? Explain your reasoning.

(2) If $a_1 = 1$, and for any positive integer $k$, $a_k - (a_{k+1} + a_{k+2})$ is still a term in the sequence.  
(i) Find the common ratio $q$.  
(ii) If $b_n = -\log_{a_{n+1}}(\sqrt{2} + 1)$, $S_n = b_1 + b_2 + \ldots + b_n$, $T_n = S_1 + S_2 + \ldots + S_n$, express $T_{2011}$ in terms of $S_{2011}$.

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