
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 $\overrightarrow{m}=( \sqrt {2}\cos \frac {x}{4},2\cos \frac {x}{4})$ and $\overrightarrow{n}=( \sqrt {2}\cos \frac {x}{4}, \sqrt {3}\sin \frac {x}{4})$, let $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.
- (I) If $f(\alpha)=2$, find the value of $\cos (\alpha+ \frac {\pi}{3})$;
- (II) In $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $(2a-b)\cos C=c\cos B$, find the range of values for $f(A)$.

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