
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 $f(x)=4\sin \omega x - \sin^{2}\left(\frac{\omega x}{2}+\frac{\pi}{4}\right) - 2\sin^{2}\omega x$ $(\omega > 0)$ is increasing on the interval $\left[-\frac{\pi}{4},\frac{3\pi}{4}\right]$ and it achieves its maximum value exactly once on the interval $[0,x]$, then the range of $\omega$ is _______.

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