
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
If the graph of the quadratic function $f(x) = ax^2 + bx + c$ ($a \neq 0$) has no intersection points with the line $y = x$, the following conclusions are drawn:
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 $x_0$, such that $f(f(x_0)) > x_0$;
4. If $a + b + c = 0$, then the inequality $f(f(x)) < x$ holds for all real numbers;
5. The graph of the function $g(x) = ax^2 - bx + c$ also definitely has no intersection points with the line $y = -x$.

Among these conclusions, the correct ones are (write down all the correct conclusion numbers).

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