
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 2, so that these $n+2$ numbers form an increasing geometric sequence. Let the product of these $n+2$ numbers be $A_n$, and let $a_n = \log_2 A_n$, where $n \in \mathbb{N}$.  
(1) Find the sum of the first $n$ terms of the sequence $\{A_n\}$, denoted as $S_n$.  
(2) Find $T_n = \tan a_2 \cdot \tan a_4 + \tan a_4 \cdot \tan a_6 + \ldots + \tan a_{2n} \cdot \tan a_{2n+2}$.

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