
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 $f(x) = A \sin(\omega x + \phi)$, where $(A > 0, \omega > 0, |\phi| < \frac{\pi}{2})$, its graph intersects the $y$-axis at $(0, \frac{3}{2})$, and its first highest and lowest points on the right side of the $y$-axis are $(x\_0, 3)$ and $(x\_0 + 2\pi, -3)$ respectively.

1. Find the analytical expression of the function $y = f(x)$.
2. How can the graph of this function be obtained through translation and scaling transformations from the graph of $y = \sin x (x \in \mathbb{R})$?
3. Find the intervals where this function is monotonically increasing and its center of symmetry.

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