
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $4\sin A\sin B-4\cos ^{2} \frac {A-B}{2}= \sqrt {2}-2$.

$(1)$ Find the magnitude of angle $C$;  
$(2)$ Given $\frac {a\sin B}{\sin A}=4$ and the area of $\triangle ABC$ is $8$, find the value of side length $c$.

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