
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 $f(x)=A\sin (\omega x+\varphi)$, where $x\in \mathbb{R}$, $A > 0$, $\omega > 0$, and $0 < \varphi < \frac{\pi}{2}$, the graph intersects with the x-axis, and the distance between two adjacent intersection points is $\frac{\pi}{2}$. Also, there is a lowest point on the graph at $M(\frac{2\pi}{3},-2)$.  
$(1)$ Find the expression for $f(x)$.  
$(2)$ After shifting the graph of $f(x)$ to the right by $\frac{\pi}{6}$ units, we get the graph of the function $y=g(x)$. When $x\in\left[\frac{\pi}{6}, \frac{2\pi}{3}\right]$, find the axis of symmetry and the symmetric point for $g(x)$.

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