
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 a function $f(x)$ on $\mathbb{R}$, for all $x \in \mathbb{R}$, satisfying $f(1-x) = f(1+x)$, $f(-x) = -f(x)$, and $f(x)$ is an increasing function on $[0, 1]$. Among the following statements, the correct ones are (Fill in all correct statement numbers):  
① $f(0) = 0$;  
② $f(x+2) = f(-x)$;  
③ $f(x)$ is an increasing function on $[-6, -4]$;  
④ $f(x)$ reaches its minimum value at $x = -1$.

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