
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 $\overrightarrow{m}=( \sqrt{3}\sin \frac{x}{4},1)$ and $\overrightarrow{n}=(\cos \frac{x}{4},\cos^2 \frac{x}{4})$, let $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.
$(1)$ If $f(x)=1$, determine the value of $\cos (x+ \frac{\pi}{3})$;
$(2)$ In an acute-angled $\triangle ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, and given $(2a-c)\cos B=b\cos C$, find the range of values for $f(2A)$.

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