
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 distance between two adjacent intersections of the graph with the x-axis is $\frac{\pi}{2}$. Also, one of the lowest points of the graph is $M\left( \frac{2\pi}{3}, -2\right)$.

1. Find the analytic expression for $f(x)$, its axis of symmetry, and the center of symmetry.
2. What transformation of the graph $y=\sin x$ results in the given graph?
3. Find the range of $f(x)$ when $x \in \left[\frac{\pi}{12}, \frac{\pi}{2}\right]$.

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