
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 $f(x)=2\sin \omega x \cdot \cos \omega x - \sqrt{3} + 2 \sqrt{3} \sin^2 \omega x$ ($\omega > 0$), its smallest positive period is $\pi$.

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

(Ⅱ) If the graph of the function $f(x)$ is translated left by $\dfrac{\pi}{6}$ units and then translated down 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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