
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $\frac{\sin A}{\sin B + \sin C} = 1 - \frac{a - b}{a - c}$.
(I) If $b = \sqrt{3}$, find the area of $\triangle ABC$ when its perimeter is at its maximum.
(II) Let $\vec{m} = (\sin A, 1)$ and $\vec{n} = (6 \cos B, \cos 2A)$, find the range of $\vec{m} \cdot \vec{n}$.

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