
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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\cos A \cos B = \sin A \sin B - \frac{\sqrt{2}}{2}$.  
$(1)$ Find the measure of angle $C$.  
$(2)$ Given $b=4$ and the area of $\triangle ABC$ is $6$, find:  
① the value of side $c$;  
② the value of $\cos \left(2B-C\right)$.

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