
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 $f(x) = \sin(\omega x + \phi)$ where $\omega > 0$ and $0 < \phi < \frac{\pi}{2}$. It is known that $f(0) = \frac{\sqrt{2}}{2}$ and for any ${x}_{1},{x}_{2} \in (\frac{\pi}{2},\pi)$, $\frac{{x}_{1}-{x}_{2}}{f({x}_{1})-f({x}_{2})} < 0({x}_{1}\neq {x}_{2})$. Determine the range of $\omega$.

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