
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 function $f(x)= \frac{ \sqrt{3}}{2}\sin 2x- \frac{1}{2}(\cos ^{2}x-\sin ^{2}x)-1 $, where $x\in\mathbb{R}$, the function $f(x)$ is translated to the left by $\frac{\pi}{6}$ units to obtain the function $g(x)$. Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively.

$(1)$ If $c=\sqrt{7}$, $f(C)=0$, and $\sin B=3\sin A$, find the values of $a$ and $b$;

$(2)$ If $g(B)=0$ and $\vec{m}=(\cos A,\cos B),\vec{n}=(1,\sin A-\cos A\tan B)$, find the range of values for $\vec{m}\cdot \vec{n}$.

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