
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 function $f(x)= \sqrt {3}\sin \frac {x}{4}\cos \frac {x}{4}+\cos ^{2} \frac {x}{4}+ \frac {1}{2}$.
$(1)$ Find the period of $f(x)$ and the center of symmetry of its graph;
$(2)$ In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $(2a-c)\cos B=b\cos C$, find the range of values for $f(A)$.

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