
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
Given the definition: If $m-\frac{1}{2} < x \leqslant m+\frac{1}{2}$ (where $m$ is an integer), then $m$ is called the nearest integer to the real number $x$, denoted as $\{x\}$, i.e., $\{x\}=m$. Based on this, the following four propositions about the function $f(x)=x-\{x\}$ are given:

1. The domain of $y=f(x)$ is $\mathbb{R}$, and the range is $\left(-\frac{1}{2},\frac{1}{2}\right]$;     
2. The smallest positive period of the function $y=f(x)$ is $1$;
3. The point $(k,0)$ is the center of symmetry of the graph of $y=f(x)$, where $k\in\mathbb{Z}$;  
4. The function $y=f(x)$ is increasing on the interval $\left(-\frac{1}{2},\frac{3}{2}\right]$.

Then, the sequence numbers of the true propositions are:

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