
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
For a function $f(x)$ defined on $\mathbb{R}$, consider the following three statements:
① If $f(1)=f(-1)$, then $f(x)$ is not an odd function;
② If $f(1)>f(-1)$, then $f(x)$ is not a monotonically decreasing function on $\mathbb{R}$;
③ If $f(1+x)=f(x-1)$ holds for all $x\in\mathbb{R}$, then $f(x)$ is a periodic function.
The correct statement number(s) are __________.

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