
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 $\overrightarrow{m}=({\sin x, 1})$ and $\overrightarrow{n}=({\sqrt{3}\cos x, \frac{1}{2}\cos 2x})$, and the function $f(x)=\overrightarrow{m} \cdot \overrightarrow{n}$.  
$(1)$ Find the maximum value of the function $f(x)$ and the corresponding values of the independent variable.  
$(2)$ In triangle $ABC$, where the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $f(A)=\frac{1}{2}$ and $a=2$, find the range of values for $b+c$.

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