
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 $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively.<br/>$(1)$ If the area of $\triangle ABC$ is $S=\frac{{a}^{2}+{c}^{2}-{b}^{2}}{4}$, find angle $B$;<br/>$(2)$ If $ac=\sqrt{3}$, $\sin A=\sqrt{3}\sin B$, and $C=\frac{π}{6}$, find $c$.

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