
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 $\overrightarrow{m} = (a+b, -c)$ and $\overrightarrow{n} = (a+b, c)$ satisfy $\overrightarrow{m} \cdot \overrightarrow{n} = (2 + \sqrt{3})ab$.

1. Find angle $C$.
2. The adjacent symmetry axes of the function $f(x) = 2\sin(A + B)\cos^2(\omega x) - \cos(A + B)\sin(2\omega x) - \frac{1}{2}\ (\omega > 0)$ are $x = x\_0$ and $x = x\_0 + \frac{\pi}{2}$, respectively. Find the interval on which $f(x)$ is increasing on $[-π, π]$.

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