
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 that the function $y=f(x)$ has a domain of $(-\infty, +\infty)$, consider the following four statements:  
① If $f(-1)=f(1)$, then $f(x)$ is an even function;  
② If $f(-1)<f(1)$, then $f(x)$ is not a decreasing function in the interval \[-2, 2\];  
③ If $f(-1) \cdot f(1)<0$, then the equation $f(x)=0$ has at least one real root in the interval $(-1, 1)$;  
④ If $|f(x)|=|f(-x)|$, $x\in\mathbb{R}$, then $f(x)$ is either an odd or an even function.  
Among these statements, the correct ones are     (fill in the numbers of all correct statements).

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