
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 arithmetic sequence $\{a_n\}$ with first term $a_1=1$, and $a_2+1$, $a_3+1$, $a_4+2$ form a geometric sequence.
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Let $b_n=\frac{2}{a_n a_{n+1}}$, find the sum of the first $n$ terms, denoted as $S_n$, for the sequence $\{b_n\}$.

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