
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 $f(x) = 2\sin\omega x \cdot \cos\omega x - \sqrt{3} + 2\sqrt{3}\sin^2\omega x$ ($\omega > 0$) has the smallest positive period of $\pi$.
(1) Find the intervals of monotonic increase for the function $f(x)$;
(2) Translate the graph of the function $f(x)$ to the left by $\frac{\pi}{6}$ units and then downward by 1 unit to obtain the graph of the function $y=g(x)$. Find the number of zeros of $y=g(x)$ in the interval $[0, 20]$.

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