
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
For the function $f(x)$ defined on $\mathbb{R}$, there are four propositions:  
① If $f(0) = 0$, then the function $f(x)$ is an odd function;  
② If $f(-4) \neq f(4)$, then the function $f(x)$ is not an even function;  
③ If $f(0) < f(4)$, then the function $f(x)$ is an increasing function on $\mathbb{R}$;  
④ If $f(0) < f(4)$, then the function $f(x)$ is not a decreasing function on $\mathbb{R}$.  
Among these propositions, the correct ones are     . (Write down the numbers of all propositions you think are correct)

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