
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 $\overrightarrow{m}=(2,-1)$ and $\overrightarrow{n}=(\sin \frac{A}{2},\cos (B+C))$, where $A$, $B$, and $C$ are the internal angles of $\triangle ABC$ corresponding to sides $a$, $b$, and $c$ respectively,

$(1)$ When $\overrightarrow{m} \cdot \overrightarrow{n}$ takes the maximum value, find the size of angle $A$;

$(2)$ Under the condition of $(1)$, when $a=\sqrt{3}$, find the range of values for $b^2+c^2$.

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