
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 $f(x)=2\cos(2x+\frac{\pi}{3})-2\cos 2x+1$.

$(1)$ Find the coordinates of the symmetry center of the graph of the function $f(x)$;

$(2)$ In an acute triangle $ABC$, the sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$, respectively, and $f(A)=0$. Find the range of values for $\frac{b}{c}$.

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