
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 triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2a^{2}\sin B\sin C=\sqrt{3}(a^{2}+b^{2}-c^{2})\sin A$. Find:

$(1)$ Angle $C$;

$(2)$ If $a=1$, $b=2$, and the midpoint of side $AB$ is $D$, find the length of $CD$.

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