
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 $f(x)=2\sin(2\omega x+\frac{π}{6})+1$.  
$(1)$ If $f(x_{1})\leq f(x)\leq f(x_{2})$ and $|x_{1}-x_{2}|_{\min }=\frac{π}{2}$, find the symmetric center of $f(x)$.  
$(2)$ Given $0 \lt \omega\ \ \lt 5$, the graph of the function $f(x)$ is shifted to the right by $\frac{π}{6}$ units to obtain the graph of the function $g(x)$. $x=\frac{π}{3}$ is a zero of $g(x)$. If the function $g(x)$ has exactly $10$ zeros in the interval $\left[m,n\right](m$,$n\in R$ and $m \lt n)$, find the smallest value of $n-m$.

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