
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]


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Let the even function $f(x)$ be defined on $(-\infty, +\infty)$ and satisfy $f(x+1) = -f(x)$, and it is an increasing function on $[-1, 0]$. Below are judgments about the function $f(x)$:  
① The graph of $f(x)$ is symmetric about the point $P\left(\frac{1}{2}, 0\right)$;  
② The graph of $f(x)$ is symmetric about the line $x=1$;  
③ $f(x)$ is an increasing function on $[0, 1]$;  
④ $f(2) = f(0)$.  
Among these judgments, the correct ones are \_\_\_\_\_\_ . (Fill in all that you think are correct)

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