
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
Let the geometric sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms given by $S_{n}=2^{n}+r$ (where $r$ is a constant). Let $b_{n}=2\left(1+\log _{2} a_{n}\right)$ for $n \in \mathbf{Z}_{+}$.
(1) Find the sum of the first $n$ terms of the sequence $\left\{a_{n} b_{n}\right\}$, denoted as $T_{n}$;
(2) If for any positive integer $n$, the following holds:
$$
\frac{1+b_{1}}{b_{1}} \cdot \frac{1+b_{2}}{b_{2}} \cdots \frac{1+b_{n}}{b_{n}} \geqslant k \sqrt{n+1},
$$
find the maximum value of the real number $k$.

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