
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 $f(x)=2\sin (\omega x+\frac{π}{6})$ where $\omega > 0$.  
$(1)$ If $f(\frac{5π}{6}+x)+f(\frac{5π}{6}-x)=0$, find the minimum value of $\omega$.  
$(2)$ If the range of $f(x)$ on the interval $[0,\frac{π}{3}]$ is $[1,2]$, find the range of values for $\omega$.

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