
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 $m=\begin{pmatrix} \sqrt{3}\sin \frac{x}{4},1\end{pmatrix}$, $n=\begin{pmatrix}\cos \frac{x}{4},\cos ^{2} \frac{x}{4}\end{pmatrix}$, define $f(x)$ as the dot product of $m$ and $n$.

(I) If $f(x)=1$, find the value of $\cos \left(x+ \frac{π}{3}\right)$;

(II) In acute triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and satisfy $(2a-c)\cdot\cos B=b\cdot\cos C$, find the range of values for $f(2A)$.

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