
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
In acute-angled triangle $\triangle ABC$, the three internal angles are $A$, $B$, and $C$, satisfying $(\sin ^{2}B+\sin ^{2}C-\sin ^{2}A)\tan A=\sin B\sin C$.  
$(1)$ Find the value of angle $A$ and the range of angle $B$.  
$(2)$ If the circumcenter of triangle $\triangle ABC$ is denoted as $O$, and $\overrightarrow{OB}•\overrightarrow{OC}=\frac{1}{2}$, find the range of $\overrightarrow{OA}•({\overrightarrow{AB}+\overrightarrow{AC}})$.

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