
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: $Max\left\{{a,b}\right\}=\left\{{\begin{array}{l}{a,a≥b,}\\{b,a＜b．}\end{array}}\right.$ Define the function $f\left(x\right)=Max\{\sin \omega x,\cos \omega x\}\left(\omega\ \ \gt 0\right)$. If the function $f\left(x\right)$ is monotonically increasing on $({\frac{π}{3},\frac{π}{2}})$, then the range of real number $\omega$ is ______.

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