
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
The function $f(x) = \lfloor x \rfloor$ gives the greatest integer less than or equal to $x$, for example, $\lfloor -3.5 \rfloor = -4$, $\lfloor 2.1 \rfloor = 2$. Consider the following statements about the function $f(x) = \lfloor x \rfloor$:   
① If $x \in [1, 2]$, then the range of $f(x)$ is $\{0, 1, 2\}$;  
② $f(x + 1) = f(x) + 1$;  
③ $f(x_1 + x_2) = f(x_1) + f(x_2)$;  
④ The function $g(x) = x - f(x)$ is periodic.  
Identify the correct statement(s) (only fill in the numbers).

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