
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 $A$, $B$, $C$ are respectively $a$, $b$, $c$. Given $a-b=2$, $c=4$, and $\sin A=2\sin B$.

$(1)$ Find the values of $a$, $b$, and $\cos B$;

$(2)$ Find the value of $\sin \left(2B-\frac{\pi}{6}\right)$.

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