
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
Insert $n$ real numbers between $1$ and $100$ so that these $n+2$ numbers form an increasing geometric sequence. Denote the product of these $n+2$ numbers as $T_n$ and let $a_n = \lg T_n$ for $n \geq 1$.

    (1) Find the general formula for the sequence $\{a_n\}$.

    (2) Let $b_n = (a_n - 1) \times 2^{n-1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted $S_n$.

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