
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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $4\sin^2\frac{{A-B}}{2}+4\sin A \sin B=2+\sqrt{2}$. Find:<br/>
$(1)$ The measure of angle $C$;<br/>
$(2)$ Given $b=4$ and the area of $\triangle ABC$ is $6$, find:<br/>
① The value of side $c$;<br/>
② The value of $\tan \left(2B-C\right)$.

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