
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 denoted as $a$, $b$, and $c$, respectively. It is given that the area of $\triangle ABC$ is $3\sqrt{15}$, $b-c=2$, and $\cos A=-\frac{1}{4}$,

(I) Find the values of $a$ and $\sin C$;

(II) Find the value of $\cos \left( 2A-\frac{\pi}{6} \right)$.

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