
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
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. The vectors $m=(a+b, -c)$ and $n=(a+b, c)$ satisfy $m \cdot n = (2+\sqrt{3})ab$.

(I) Find angle $C$;

(II) The function $f(x) = 2\sin(A + B)\cos^2x - \cos(A + B)\sin 2x - \frac{1}{2}$. Find the interval(s) where $f(x)$ is increasing on $R$.

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