
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 $f(x)=2\sin \omega x \sin(\omega x+\frac{\pi}{3})+\cos 2\omega x-\frac{1}{2}$, if the function $f(x)$ has exactly two zeros in the interval $\left[0,\pi \right]$, determine the range of real number $\omega$.

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