
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 \( f(x)=\frac{\sqrt{3}}{2} \sin 2 x-\frac{1}{2}\left(\cos ^{2} x-\sin ^{2} x\right)-1, x \in \mathbf{R} \), the function \( g(x) \) is obtained by shifting \( f(x) \) to the left by \(\frac{\pi}{6}\) units. In triangle \( \triangle ABC \), the sides opposite to angles \( A, B, \) and \( C \) are \( a, b, \) and \( c \), respectively.
(1) Given \( c=\sqrt{7} \), \( f(C)=0 \), and \( \sin B=3 \sin A \), find the values of \( a \) and \( b \).
(2) Given \( g(B)=0 \), \( \vec{m}=(\cos A, \cos B) \), and \( \vec{n}=(1, \sin A-\cos A \tan B)\), determine the range of values for \( \vec{m} \cdot \vec{n} \).

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