
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 interval of monotonic increase for the function $f(x)$;  
(2) The graph of $f(x)$ is obtained by translating the graph of $y=\sin x$ in what way;  
(3) If the graph of the function $f(x)$ is translated to the left by $\frac{\pi}{6}$ units and then translated up by 1 unit to get the graph of the function $y=g(x)$, and 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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