
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
Define an even function $f(x)$ on $(-\infty, +\infty)$ that satisfies $f(x+1) = -f(x)$, and is increasing on $[-1, 0]$. Below are judgments about $f(x)$:
① $f(x)$ is a periodic function;
② The graph of $f(x)$ is symmetric about the line $x=1$;
③ $f(x)$ is increasing on $[0, 1]$;
④ $f(2) = f(0)$.
Among these judgments, fill in the correct ones.

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