
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
Given the function $f(x)=\sqrt{3}\sin\frac{x}{2}\cos\frac{x}{2}-\cos^{2}\frac{2x}{2}+\frac{1}{2}$.

(1) Find the interval(s) where the function $f(x)$ is monotonically decreasing.

(2) In triangle $ABC$, if angle $A$, $B$, and $C$ correspond to sides $a$, $b$, and $c$ respectively, and $f(A)=\frac{1}{2}$, $a=\sqrt{3}$, and $\sin B=2\sin C$, find $c$.

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