
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 all $x \in \mathbb{R}$, the equation $f(x+6)=f(x)+f(3)$ holds. Consider the following statements:  
① $f(3)=0$;  
② $f(-3)=0$;  
③ The line $x=6$ is an axis of symmetry for the graph of the function $y=f(x)$;  
④ The function $y=f(x)$ is increasing on the interval $[-9, -6]$.  
Among these, 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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