
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 the function $f(x)=\cos(x+ \frac{\pi}{6})$. Consider the following statements:  
$(1)$ A period of $f(x)$ is $-2\pi$;  
$(2)$ The graph of $f(x)$ is symmetric about the line $x= \frac{5\pi}{6}$;  
$(3)$ A zero of $f(x+\pi)$ is $x= \frac{\pi}{3}$;  
$(4)$ $f(x)$ is monotonically decreasing in the interval $\left( \frac{\pi}{2},\pi\right)$,  
Among these statements, the correct ones are __________ (Fill in the numbers of all correct statements).

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