
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
Let $\lfloor x \rfloor$ represent the greatest integer less than or equal to $x$, for example, $\lfloor 3.1 \rfloor = 3$, $\lfloor -3.4 \rfloor = -4$, $\lfloor 0 \rfloor = 0$. Define the function $f(x) = \lfloor x \rfloor - x$, where $x \in \mathbb{R}$. Consider the following four statements about the function $f(x)$:
1. The range of $f(x)$ is $[0, 1)$.
2. $f(x)$ is an even function.
3. $f(x)$ is a periodic function, with the smallest positive period being 1.
4. $f(x)$ is an increasing function.
Identify the correct statements.

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