
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 that the minimum positive period of the function $f(x)=(\sin \omega x+\cos \omega x)^{2}+2\cos ^{2}\omega x\ (\omega > 0)$ is $\frac{2\pi}{3}$.
(I) Find the value of $\omega$;
(II) 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(s) where $y=g(x)$ is increasing.

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