
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 vectors $m=\left(\begin{matrix} \sqrt{3}\sin \frac{x}{4},1 \end{matrix}\right)$, $n=\left(\begin{matrix} \cos \frac{x}{4},\cos ^{2} \frac{x}{4} \end{matrix}\right)$.

$(1)$ If $m \perp n$, find the value of $\cos \left(\frac{2\pi}{3}-x\right)$;

$(2)$ Let $f(x)=m \cdot n$, in $\triangle ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and satisfy $(2a-c)\cos B=b\cos C$, find the range of the function $f(A)$.

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