
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 $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$, where $\overrightarrow{a}=(2\cos x,\sqrt{3}\sin 2x)$ and $\overrightarrow{b}=(\cos x,1)$, $x\in \mathbb{R}$.

$(1)$ Find the intervals of monotonic increase for the function $y=f(x)$;

$(2)$ In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, $f(A)=2$, $a=\sqrt{7}$, and $\sin B=2\sin C$. Find the area of $\triangle ABC$.

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