
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 $\mathbb{R}$ that satisfies: $f(x+1) = -f(x)$, and it is an increasing function on the interval $[0, 1]$. Regarding $f(x)$, the following judgments are made:
① $f(x)$ is a periodic function;
② The graph of $f(x)$ is symmetric about the line $x=1$;
③ $f(x)$ is a decreasing function on the interval $[1, 2]$;
④ $f(x)$ is a decreasing function on the interval $[-2, 0]$.
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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