
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 an increasing arithmetic sequence $\{a_n\}$, where $a_1=2$, $a_3$ is the geometric mean of $a_1$ and $a_9$.  
(1) Find the common formula for the sequence $\{a_n\}$.  
(2) If $b_n= \frac{1}{(n+1)a_n}$, $S_n$ is the sum of the first $n$ terms of the sequence $\{b_n\}$, find the value of $S_{100}$.

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