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

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