
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
Let the function $f(x) = (\sin \omega x + \cos \omega x)^2 + 2\cos^2 \omega x$ ($\omega > 0$) have the smallest positive period of $\frac{2\pi}{3}$.

(1) Find the value of $\omega$;  
(2) Find the range of the function $f(x)$ on $[0, \frac{\pi}{3}]$;  
(3) If the graph of the function $y = g(x)$ is obtained by shifting the graph of $y = f(x)$ to the right by $\frac{\pi}{2}$ units, find the interval of increase for $y = g(x)$.

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