
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 $y=|x|$ and the function $y=(\sqrt{x})^{2}$ represent the same function;  
② The graph of an odd 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 $[0,4]$;  
④ The graph of the function $y=3(x-1)^2$ can be obtained by shifting the graph of $y=3x^2$ to the right by one unit;  
⑤ Suppose the function $f(x)$ is continuous on the interval $[a,b]$ and $f(a) \cdot f(b) < 0$, then the equation $f(x)=0$ has at least one real root in the interval $[a,b]$;  
Among these, the correct propositions are \_\_\_\_\_\_. (Fill in the numbers of all correct propositions)

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