
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 vectors $a=(\sin \frac{\omega}{2}x,\sin \omega x)$ and $b=(\sin \frac{\omega}{2}x,\frac{1}{2})$, where $\omega > 0$, if the function $f(x)=a\cdot b-\frac{1}{2}$ has no zeros in the interval $(\pi,2\pi)$, determine the range of $\omega$.

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