
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
Consider the following five propositions:  
① If the equation $x^2+(a-3)x+a=0$ has one positive real root and one negative real root, then $a<0$;  
② The function $y= \sqrt {x^{2}-1}+ \sqrt {1-x^{2}}$ is an even function, but not an odd function;  
③ If the range of the function $f(x)$ is $[-2, 2]$, then the range of the function $f(x+1)$ is $[-3, 1]$;  
④ Given that the domain of the function $y=f(x)$ is $\mathbb{R}$, the graph of the function $y=f(1-x)$ is symmetrical to the graph of $y=f(x-1)$ about the $y$-axis;  
⑤ For the curve  

$$
y= 
\begin{cases} 
3-x^{2} & (x\in[- \sqrt {3}, \sqrt {3}]) \\
x^{2}-3 & (x\in(-\infty, - \sqrt {3})\cup( \sqrt {3}, +\infty)) 
\end{cases}
$$

and the line $y=a$ ($a\in\mathbb{R}$), if the number of common points is $m$, then the value of $m$ cannot be 1.  
The correct proposition numbers are ________ (list all correct proposition numbers).

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