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

(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.

(Ⅱ) In an acute-angled triangle $ABC$, with the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively, and satisfying the condition $(2a-c)\cos B = b\cos C$, find the range of values for $f(2A)$.

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