
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 $f(x) = \sin x (\sqrt{3}\cos x + \sin x) + \frac{1}{2}$.

(I) If $x \in [0, \pi]$, find the increasing interval of $f(x)$.

(II) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ with opposite sides $a$, $b$, $c$ respectively, and $c = \sqrt{3}$, $f(C) = 2$, $\sin B = 2\sin A$. Find the area of $\triangle ABC$.

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