
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 $f(x)=2\sqrt{3}\sin\frac{\omega x}{2}\cos\frac{\omega x}{2}+6\cos^2\frac{\omega x}{2}-3$ ($\omega>0$)  
(1) If $y=f(x+\theta)$ ($0<\theta<\frac{\pi}{2}$) is an even function with the smallest positive period of $\pi$, find the values of $\omega$ and $\theta$;  
(2) If $g(x)=f(3x)$ is an increasing function on $(0, \frac{\pi}{3})$, find the maximum value of $\omega$.

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