
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 $f(x)=4\sin \omega x\cdot \sin ^{2}\left( \dfrac {\pi}{4}+ \dfrac {\omega x}{2}\right)+\cos 2\omega x$, determine the range of $\omega$ if the function is increasing on the interval $\left[- \dfrac {\pi}{2}, \dfrac {2\pi}{3}\right]$.

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