
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
In the geometric sequence $\{a_n\}$, $a_1=1$, and $a_2$ is the arithmetic mean of $a_1$ and $a_3-1$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) If the sequence $\{b_n\}$ satisfies $b_n= \frac {n(n+1)a_n+1}{n(n+1)}$ $(n\in\mathbb{N}^*)$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, $S_n$.

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