
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 that in triangle $\triangle ABC$, the lengths of the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively, and the radius of the circumcircle of $\triangle ABC$ is $\sqrt{3}$, and $\sin ^{2}B+\sin ^{2}C-\sin B\sin C=\sin ^{2}A$. Find:<br/>
$(1)$ the length of side $a$;<br/>
$(2)$ the range of values for the inradius $r$ of $\triangle ABC$.

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