
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 $\dfrac{2\pi}{3}$.

(1) Find the value of $\omega$;

(2) If the graph of the function $y=g(x)$ is obtained by shifting the graph of $y=f(x)$ to the right by $\dfrac{\pi}{2}$ units, find the intervals of increase for $y=g(x)$.

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