
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 $f(x)=\sin({ωx+φ})({ω＞0,|φ|＜\frac{π}{2}})$. If $f({\frac{π}{6}+x})=-f({\frac{π}{6}-x})$ and $f({-\frac{{5π}}{{24}}+x})=f({-\frac{{5π}}{{24}}-x})$, and $f\left(x\right)$ is monotonically increasing on the interval $({\frac{π}{3}},{\frac{π}{2}})$, determine the possible values of $\omega$.

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