
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 denoted as $S_{n}$, satisfying $S_{n} = 2^{n} + r$ (where $r$ is a constant). Define $b_{n} = 2\left(1 + \log_{2} a_{n}\right)$ for $n \in \mathbf{N}^{*}$.

(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 $\frac{1 + b_{1}}{b_{1}} \cdot \frac{1 + b_{2}}{b_{2}} \cdots \frac{1 + b_{n}}{b_{n}} \geq k \sqrt{n + 1}$ holds, determine the maximum value of the real number $k$.

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