
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 $\overrightarrow{m}=(\frac{1}{2}\sin x,\frac{\sqrt{3}}{2})$ and $\overrightarrow{n}=(\cos x,\cos^2 x-\frac{1}{2})$ where $x\in\mathbb{R}$, and the function $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$.

$(1)$ Find the equation of the axis of symmetry for $f(x)$;

$(2)$ In an acute triangle $ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $f(A)=0$ and $a=\sqrt{3}$, find the range of values for $b+c$.

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