
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
Regarding the function $f(x)=4\sin (2x+ \frac {\pi}{3})(x\in\mathbb{R})$, the following propositions are given:  
$(1)$ From $f(x_{1})=f(x_{2})=0$, it can be concluded that $x_{1}-x_{2}$ must be an integer multiple of $\pi$;  
$(2)$ The expression of $y=f(x)$ can be rewritten as $y=4\cos (2x- \frac {\pi}{6})$;  
$(3)$ The graph of $y=f(x)$ is symmetric about the point $(- \frac {\pi}{6},0)$;  
$(4)$ The graph of $y=f(x)$ is symmetric about the line $x=- \frac {\pi}{6}$.  
Among these, the correct proposition numbers are __________.

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