
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
The function $f(x) = \sin(\omega x + \phi)$ ($\omega > 0$, $0 < \phi < \frac{\pi}{2}$) satisfies $f(0) = \frac{\sqrt{2}}{2}$, and for any $x_1, x_2 \in \left(\frac{\pi}{2}, \pi\right)$ with $x_1 \neq x_2$, we have $\frac{x_1 - x_2}{f(x_1) - f(x_2)} < 0$. Determine the range of possible values for $\omega$.

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