
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
Let two functions $f(x)$ and $g(x)$ be defined on domain $D$, with their ranges being $[a, b]$ and $[c, d]$ respectively. There are the following four propositions:  
① "a > d" is the necessary and sufficient condition for "$f(x_1) > g(x_2)$ for any $x_1, x_2 \in D$ to always hold";  
② "a > d" is the sufficient but not necessary condition for "$f(x_1) > g(x_2)$ for any $x_1, x_2 \in D$ to always hold";  
③ "a > d" is the necessary and sufficient condition for "$f(x) > g(x)$ for any $x \in D$ to always hold";  
④ "a > d" is the sufficient but not necessary condition for "$f(x) > g(x)$ for any $x \in D$ to always hold".  
Among these propositions, the correct ones are (please list all correct proposition numbers).

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