
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 that the common ratio of a geometric sequence $\{a_{n}\}$ is $q\left(q\neq 1\right)$, $a_{1}=2$, $2a_{1}+a_{3}=3a_{2}$, and the common difference of an arithmetic sequence $\{b_{n}\}$ is $-1$, $b_{3}=2$. Let $\min \{x_{1}$, $x_{2}$, $\ldots $, $x_{s}\}$ represent the smallest number among $x_{1}$, $x_{2}$, $\ldots $, $x_{3}$ of $s$ numbers.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Let $c_{n}=\min \{\frac{2^nb_1}{a_1}$, $\frac{2^nb_2}{a_2}$, $\frac{2^nb_3}{a_3}$, $\ldots $, $\frac{2^nb_n}{a_n}\}$. 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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