
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
Due to the outstanding contributions of the famous German mathematician Dirichlet in number theory, mathematical analysis, and physics, the function $D(x)= \begin{cases} 0, & \text{if } x \text{ is irrational} \\ 1, & \text{if } x \text{ is rational} \end{cases}$ is named the Dirichlet function. Given the function $f(x)=x-D(x)$, among the following statements:  
$(1)$ The domain and range of the function $f(x)$ are both $\mathbb{R}$;  
$(2)$ The function $f(x)$ is an odd function;  
$(3)$ The function $f(x)$ is a periodic function;  
$(4)$ The function $f(x)$ is a monotonic function in the interval $[2,3]$.  
The correct conclusion is $\boxed{\text{(1)}}$.

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