
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 $f(x)=\frac{1}{2}\sin \omega x-\frac{\sqrt{3}}{2}\cos \omega x$ ($\omega > 0$), whose zeros form an arithmetic progression with a common difference of $\frac{\pi}{2}$. If $f(x)$ is monotonically increasing on the interval $[0,\alpha]$, determine the possible range of $\alpha$.

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