
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 $a$, $b$, $c$ respectively, and $a=2b$. Also, $\sin A$, $\sin C$, $\sin B$ form an arithmetic sequence.

(I) Find the value of $\cos (B+C)$;

(II) If ${{S}_{\triangle ABC}}=\frac{3\sqrt{15}}{3}$, find the value of $c$.

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