
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 $$f(x) = \sin(2x + \frac{\pi}{6}) + \cos(2x - \frac{\pi}{3})$$,
(I) Find the maximum value of $f(x)$ and the value of $x$ at which it attains its maximum;
(II) In triangle $ABC$, let the sides opposite angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. If $f(C) = 1$ and $c = 2\sqrt{3}$, with $\sin A = 2 \sin B$, find the area of triangle $ABC$.

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