
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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Given an even function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(x+2) = f(x) + f(1)$ and is monotonically increasing in the interval $[0, 1]$, then, among the following statements about the properties of this function $f(x)$:
① The graph of the function $y=f(x)$ is symmetric about the line $x=1$;  
② The function $y=f(x)$ is a periodic function;  
③ For $x \in [-3, -2]$, $f'(x) \geq 0$;  
④ The points on the graph of the function $y=f(x)$ with even $x$-coordinates are all local minima of the function.  
The correct statements are.

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