
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
Consider the circle described by the equation $x^2 + y^2 + 2ax - 2ay = 0$. Among the following statements:  
① It is symmetric about the line $x+y=0$;  
② Its center lies on the x-axis;  
③ It passes through the origin;  
④ Its radius is $\sqrt{2}a$.  
Identify the correct statements. (Provide the numbers of all correct statements)

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