
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 a geometric sequence $\{a_n\}$ with the first term $a_1 = \frac{1}{2}$ and the sum of the first $n$ terms denoted by $S_n$, where the common ratio is not $1$, and $a_4 + S_4$, $a_5 + S_5$, and $a_6 + S_6$ form an arithmetic sequence.

(1) Find the general formula for the $n$-th term of the geometric sequence $\{a_n\}$ and the sum of the first $n$ terms $S_n$.

(2) For $n \in \mathbb{N}^*$, insert $3^n$ numbers between $a_n$ and $a_{n+1}$ to form an arithmetic sequence with these $3^n + 2$ numbers. Let the sum of the $3^n$ inserted numbers be $b_n$. Calculate the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted by $T_n$.

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