
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
For the function $f(x)=4\sin \left(2x+\frac{\pi }{3}\right)$, the following propositions are given:

$(1)$ From $f(x_1)=f(x_2)$, it can be concluded that $x_1-x_2$ is an integer multiple of $\pi$; $(2)$ The expression for $f(x)$ can be rewritten as $f(x)=4\cos \left(2x-\frac{\pi }{6}\right)$; $(3)$ The graph of $f(x)$ is symmetric about the point $\left(-\frac{\pi }{6},0\right)$; $(4)$ The graph of $f(x)$ is symmetric about the line $x=-\frac{\pi }{6}$; Among these, the correct propositions are ______________.

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