
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
In triangle $\triangle ABC$, the sides opposite the three interior angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Please choose any one of the following three conditions:①$\left(2c-a\right)\cos B=b\cos A$; ②$a^{2}+c^{2}-b^{2}=\frac{4\sqrt{3}}{3}S_{\triangle ABC}$; ③$2b\sin (A+\frac{π}{6})=a+c$, and complete the following questions:<br/>$(1)$ If $3a+b=2c$, find $\cos C$;<br/>$(2)$ If $b=2$, and $\frac{1}{\sin A}+\frac{1}{\sin C}=\frac{4\sqrt{3}}{3}$, find the area of $\triangle ABC$.

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