
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
Let $f(x) = 3\sin(wx + \varphi)$ $(w > 0, -\frac{\pi}{2} < \varphi < \frac{\pi}{2})$ be a function whose graph is symmetric about the line $x = \frac{2\pi}{3}$, and its period is $\pi$. Then, among the following statements, $(1) f(x)$'s graph passes through the point $(0, \frac{1}{2})$; $(2) f(x)$ has a center of symmetry at $(\frac{5\pi}{12}, 0)$; $(3) f(x)$ is a decreasing function on the interval $\left[\frac{\pi}{12}, \frac{2\pi}{3}\right]$; $(4)$ shifting the graph of $f(x)$ to the right by $|\varphi|$ units results in the graph of the function $y = 3\sin wx$. The correct statement(s) is/are __________.

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