
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 an even function $f(x)$ defined on $\mathbb{R}$ that satisfies the condition: $f(x+1) = -f(x)$, and it is an increasing function on the interval $[-1, 0]$. Consider the following statements about $f(x)$:  
① $f(x)$ is a periodic function;  
② $f(x)$ is an increasing function on the interval $[0, 1]$;  
③ $f(x)$ is a decreasing function on the interval $[1, 2]$;  
④ $f(2) = f(0)$.  
Among these statements, the correct ones are. (Note: Fill in all the correct statement numbers.)

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