
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 $f(x) = x$ has no real roots, then among the following statements:  
(1) The equation $f[f(x)] = x$ definitely has no real roots;  
(2) If $a > 0$, then the inequality $f[f(x)] > x$ holds for all real numbers $x$;  
(3) If $a < 0$, then there must exist a real number $x$, such that $f[f(x)] > x$;  
(4) If $a + b + c = 0$, then the inequality $f[f(x)] < x$ holds for all $x$.  
Among these, the correct statement numbers are (write down all the numbers of true statements).

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