
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 sum of the first \(n\) terms \(S_n\) such that \(S_n = 2^n + r\) (where \(r\) is a constant), let \(b_n = 2(1 + \log_2 a_n)\) for \(n \in \mathbb{N}^*\).

1. Find the sum of the first \(n\) terms of the sequence \(\{a_n b_n\}\), denoted as \(T_n\).
2. If for any positive integer \(n\), the inequality \(\frac{1 + b_1}{b_1} \cdot \frac{1 + b_2}{b_2} \cdots \cdot \frac{1 + b_n}{b_n} \geq k \sqrt{n + 1}\) holds, determine \(k\).

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