
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
Consider the geometric sequence $\{a_n\}$, where $a_1$ is given, and $a_2$ is the arithmetic mean of $a_1$ and $a_3 - 1$.
If the sequence $\{b_n\}$ satisfies
$$
b_n = \frac{n(n+1)a_n}{n(n+1)} \quad (n \in \mathbb{N}^{*}),
$$
find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $S_n$.

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