
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 two positive sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfying $(a_{n}-b_{n})b_{n}=1$ and $\frac{1}{{a}_{n}}=\frac{{b}_{n}}{{n}^{2}+1}$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ If the sequence $\{c_{n}\}$ satisfies $c_{n}=[a_{n}+a_{n+1}]+b_{n}$, where $\left[x\right]$ denotes the greatest integer not exceeding $x$, find the sum of the first $n$ terms of $\{c_{n}\}$ denoted by $S_{n}$.

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