
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 $m=(2\sqrt{3}\sin\frac{x}{4}, 2)$ and $n=(\cos\frac{x}{4}, \cos^2\frac{x}{4})$,

(I) If $m\cdot n=2$, find the value of $\cos(x+\frac{\pi}{3})$;

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

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