
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
Among the following propositions, the correct one(s) is(are) ______ (write down the number(s) of the correct proposition(s)):
(1) There exists $x_0 \in [a, b]$ such that $f(x_0) > g(x_0)$, it is only necessary that $\max{f(x)} > \min{g(x)}$;
(2) For all $x \in [a, b]$, $f(x) > g(x)$ holds true, it is only necessary that $\min{[f(x) - g(x)]} > 0$;
(3) For all $x_1 \in [a, b]$ and $x_2 \in [c, d]$, $f(x_1) > g(x_2)$ holds true, it is only necessary that $\min{f(x)} > \max{g(x)}$;
(4) There exists $x_1 \in [a, b]$ and $x_2 \in [c, d]$ such that $f(x_1) > g(x_2)$, it is only necessary that $\min{f(x)} > \min{g(x)}$.

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