
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 $f(x) = ax^2 + bx + c$ ($a \neq 0$), and the equation $f(x) = x$ has no real roots, consider the following statements:  
① The equation $f[f(x)] = x$ also has no real roots;  
② If $a < 0$, then there must exist a real number $x$ such that $f[f(x)] > x$;  
③ If $a > 0$, then the inequality $f[f(x)] > x$ holds for all real numbers $x$;  
④ If $a + b + c = 0$, then the inequality $f[f(x)] < x$ holds for all real numbers $x$;  
Among the above statements, the correct ones are: (Fill in all the correct statement numbers).

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