
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 $f\left(x\right)=\sin \left(\omega x+\varphi \right)$ is monotonic on the interval $({\frac{π}{6},\frac{π}{2}})$, where $\omega$ is a positive integer, $|\varphi | \lt \frac{π}{2}$, and $f({\frac{π}{2}})=f({\frac{{2π}}{3}})$.  
$(1)$ Find the axis of symmetry of the graph of $y=f\left(x\right)$;  
$(2)$ If $f({\frac{π}{6}})=\frac{{\sqrt{3}}}{2}$, find $\varphi$.

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