
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 propositions, the correct ones are:  
① If $A=\{x|x>0\}$, $B=\mathbb{R}$, then the function $f: x \rightarrow y=x^2$ is a mapping from $A$ to $B$;  
② Given a function $f(x)$ such that for any real numbers $x$, $y$, $f(x+y)=f(x) \cdot f(y)$, and $f(1) \neq 0$, then $f(0)=1$;  
③ There are infinitely many functions that are both odd and even;  
④ If $f(x)$ is an even function defined on $\mathbb{R}$, then $f(x) \cdot f(-x) > 0$;  
⑤ If there exists a constant $M$ such that for any $x$ in the domain of the function $y=f(x)$, $f(x) \leq M$, then $M$ is the maximum value of $y=f(x)$.

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