
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 that the function $y=f(x)$ is an even function on $\mathbb{R}$, and for any $x \in \mathbb{R}$, the equation $f(x+6)=f(x)+f(3)$ holds. When $x_{1}, x_{2} \in [0,3]$ and $x_{1} \neq x_{2}$, it is always true that $\dfrac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 0$. Consider the following statements:  
$(1) f(3)=0$;  
$(2)$ The line $x=-6$ is an axis of symmetry for the graph of the function $y=f(x)$;  
$(3)$ The function $y=f(x)$ is increasing on $[-9,-6]$;  
$(4)$ The function $y=f(x)$ has four zeros on $[-9,9]$.  
The correct statement(s) is/are \_\_\_\_\_\_ (Fill in all correct statement numbers).

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