
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 $f(x)=\cos ^{2}\omega x+2\sin \omega x\cos \omega x-\sin ^{2}\omega x$ ($\omega\ \gt 0$), which has a maximum value and no minimum value in the interval $\left(\frac{π}{{12}},\frac{π}{3}\right)$, find the range of values for $\omega$.

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