
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 $f(x) = \cos 2x + 2\sqrt{3}\sin x \cos x$, where $x \in \mathbb{R}$.

(I) Find the intervals of monotonic increase for the function $f(x)$.

(II) In $\triangle ABC$, where angles $A$, $B$, and $C$ are opposite the sides $a$, $b$, and $c$ respectively, if $f(A) = 1$, $a = \sqrt{3}$, and $b + c = 3$, find the area of $\triangle ABC$.

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