
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 the function $f(x) = |x^2 - 2ax + b|$ ($x \in \mathbb{R}$), consider the following four statements:  
① $f(x)$ is necessarily an even function;  
② When $f(0) = f(2)$, the graph of $f(x)$ must be symmetric about $x=1$;  
③ If $a^2 - b \leq 0$, then $f(x)$ is an increasing function on the interval $[a, +\infty)$;  
④ $f(x)$ has a maximum value of $|a^2 - b|$.  
Among these statements, the correct ones are.

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