
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 the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfy: $a_{1}=2$, $b_{1}=1$, $a_{n+1}=\frac{3}{4}a_{n}+\frac{1}{4}b_{n}$, $b_{n+1}=\frac{3}{4}b_{n}+\frac{1}{4}a_{n}$, where $n\in N^{*}$.  
$(1)$ Find the general formula for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ If $c_{n}=3a_{n}+b_{n}$, find the sum of the first $n$ terms of the sequence $\{nc_{n}\}$ denoted by $S_{n}$.

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