
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 a sequence $\{a_{n}\}$, where $a_{1}=1$, and for any $i\in N^{*}$, $a_{n+i}-a_{n}=i$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ The sequence $\{b_{n}\}$ satisfies: $\frac{b_{n+1}}{b_{n}}=\frac{a_{n}}{a_{n+2}}$, and $b_{1}=1$. 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}
user