
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
Among the following four propositions, the correct ones are \_\_\_\_\_\_ (write down the serial numbers of all correct propositions)  
① If the domain of the function $f(x)$ is $[0,2]$, then the domain of the function $f(2x)$ is $[0,4]$;  
② Let set $A=\{-1,0,1\}$, $B=\{-1,1\}$, then among all mappings from $A$ to $B$, there are 4 even functions;  
③ There does not exist a real number $a$ such that the range of the function $$f(x)=\pi^{ax^{2}+2ax+3}$$ is $(0,1]$  
④ If the function $$f(x)=\log_{\frac{1}{2}}(x^{2}-ax+3a)$$ is decreasing on $[2,+\infty)$, then $-4<a\leq4$.

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