
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 $f(x)=2\sin x\cos x+2\sqrt{3}\cos^2 x-\sqrt{3}$.

(1) Find the smallest positive period and the interval where the function is monotonically decreasing.

(2) In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, where $a=7$. If acute angle $A$ satisfies $f(\frac{A}{2}-\frac{\pi}{6})=\sqrt{3}$, and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of triangle $ABC$.

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