
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
Let the function $f(x) = \vec{a} \cdot \vec{b}$, where $\vec{a} = (2\cos x, \sqrt{3}\sin 2x)$ and $\vec{b} = (\cos x, 1)$, and $x \in \mathbb{R}$.
(1) Find the period and the interval of monotonous increase for the function $y=f(x)$;
(2) In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $f(A) = 2$, $a = \sqrt{7}$, 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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