
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 $\lfloor x \rfloor$ and $\{x\}$ as functions defined for all real numbers, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\{x\}$ is the fractional part of $x$, i.e., $x - \lfloor x \rfloor$. Then, among the following statements, fill in the correct statement number(s).  
① $\lfloor -x \rfloor = -\lfloor x \rfloor$;  
② $\lfloor x \rfloor + \lfloor y \rfloor \leq \lfloor x + y \rfloor$;  
③ $\{x\} + \{y\} \geq \{x + y\}$;  
④ $\{x\}$ is a periodic function.

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