
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 such that these \( n+2 \) numbers form an increasing geometric sequence. Let the product of these \( n+2 \) numbers be denoted as \( T_n \). Furthermore, let \( a_n = \lg T_n \) for \( n \geq 1 \). Define \( b_n = \tan a_n \cdot \tan a_{n+1} \). Find the sum of the first \( n \) terms of the sequence \(\{b_n\}\) denoted as \( S_n \).

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