
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
An even function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+1) = -f(x)$, and it is increasing on the interval $[-1, 0]$. Consider the following statements about $f(x)$:  
1. $f(x)$ is a periodic function;  
2. $f(x)$ is increasing on the interval $[0, 1]$;  
3. $f(x)$ is decreasing on the interval $[1, 2]$;  
4. $f(x)$ is symmetric about the line $x=1$.  
List the correct statement numbers. (Write down all correct statement numbers).

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