
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 $\overrightarrow{m}=( \sqrt{3}\sin \frac{x}{4},1)$, $\overrightarrow{n}=(\cos \frac{x}{4},\cos^2 \frac{x}{4})$, let $f(x)= \overrightarrow{m}· \overrightarrow{n}$.

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

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

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