
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 following four propositions:
① The function $f(x)$ is increasing when $x>0$ and also increasing when $x<0$, so $f(x)$ is an increasing function;
② The graph of a direct proportion function must pass through the origin of the Cartesian coordinate system;
③ If the domain of the function $f(x)$ is $[0,2]$, then the domain of the function $f(2x)$ is $[1,2]$;
④ The increasing interval of $y=x^2-2|x|-3$ is $[1,+\infty)$.
Among these, the correct proposition numbers are. (Fill in all the correct proposition numbers)

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