
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 $a_{1}=1$, $a_{2}=2$, and $a_{n+2}=a_{n}+2\times 3^{n}$ for $n\in N^{*}$, and $b_{n}=a_{n}+a_{n+1}$ for $n\in N^{*}$.  
$(1)$ Find the general formula for the sequence $\{b_{n}\}$.  
$(2)$ If $b_{n}c_{n}=\frac{{4(n+1)}}{{4n^2-1}}$ for $n\in N^{*}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$.

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