
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 $f(x) = \sin^2 \omega x + 2 \sqrt{3} \cos \omega x \sin \omega x + \sin(\omega x + \frac{\pi}{4})\sin(\omega x - \frac{\pi}{4})$ ($\omega > 0$), and the smallest positive period of $f(x)$ is $\pi$.
1. Find the value of $\omega$;
2. Find the intervals of monotonic increase of $f(x)$ in the interval $(0, \pi)$.

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