
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 triangle $\triangle ABC$ with angles $A$, $B$, $C$ and corresponding sides $a$, $b$, $c$. It is known that $a^{2}=c^{2}+b^{2}-cb$ and $a=\sqrt{3}$. Also, an arithmetic sequence $\{a_{n}\}$ with common difference $d=\frac{a}{\sin A}$ and first term $a_{1}=2$.

(I) Find angle $A$ and the general term formula for the sequence $\{a_{n}\}$;

(II) If a sequence $\{b_{n}\}$ satisfies $\log_{3}{b_{n}}=\frac{a_{n}}{2}$, find the sum of the first $n$ terms ($S_{n}$) of the sequence $\{\frac{a_{n}b_{n}}{2}\}$.

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