
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 $f(x)=2\sin \omega x\cos \omega x+2 \sqrt{3}\sin ^{2}\omega x- \sqrt{3} (\omega > 0)$ has the smallest positive period of $\pi$.  
$(1)$ Find the intervals of increase for the function $f(x)$;  
$(2)$ Shift the graph of the function $f(x)$ to the left by $\dfrac{\pi}{6}$ units and then upward by $1$ unit to obtain the graph of the function $y=g(x)$. If $y=g(x)$ has at least $10$ zeros in the interval $[0,b] (b > 0)$, find the minimum value of $b$.

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