
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 acute triangle $\triangle ABC$, the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively. Given that $a\sin\left(C+\frac{\pi}{3}\right)=b\sin\frac{\pi}{3}$. Find:<br/>
$(1)$ $\tan\frac{A}{4}$;<br/>
$(2)$ the range of $\frac{{\cos B - \cos C}}{{\sin B + \sin C}}$.

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