
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 $x \in \mathbb{R}$, $A > 0$, $\omega > 0$, $0 < \varphi < \frac{\pi}{2}$, 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 $M\left( \frac{2\pi}{3}, -2\right)$.

(1) Find the values of $A$, $\omega$, and $\varphi$.  
(2) Describe the symmetry center of the graph of the function $f(x)$ and the interval where it is monotonically increasing.  
(3) When $x \in \left[ \frac{\pi}{12}, \frac{\pi}{2}\right]$, find the range of $f(x)$.

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