
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
Let $a_{1}=2$, $a_{n+1}=\frac{1}{2a_{n}+1}$, $b_{n}=|\frac{2a_{n}+2}{2a_{n}-1}|$, $n\in N^{*}$.  
$(1)$ Find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ If the sequence $c_{n}=\frac{n+2}{n(n+1)b_{n}}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$.

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