
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
For the geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. It is known that for any $n \in \mathbb{N}^*$, the point $(n, S_n)$ is on the graph of the function $y = b^x + r$ (where $b>0$ and $b \neq 1$, and $b$, $r$ are constants).

(1) Find the value of $r$;

(2) When $b = 2$, let $b_n = \frac{n+1}{4a_n}$ for $n \in \mathbb{N}^*$. Find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted $T_n$.

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