
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 $(-\infty, +\infty)$ that satisfies $f(x+2) = -f(x)$, and it is increasing on $[-2,0]$, consider the following statements about $f(x)$:
(1) $f(x)$ is a periodic function;
(2) $f(x)$ is increasing on $[0,2]$;
(3) $f(x)$ is decreasing on $[2,4]$;  
(4) The graph of $f(x)$ is symmetric about the line $x=2$.
Determine the correct statement(s).

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