
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 famous German mathematician Dirichlet made significant achievements in the field of mathematics. He was the first person in the history of mathematics to pay attention to concepts and consciously "replace intuition with concepts." The function named after him, $D\left(x\right)=\left\{\begin{array}{l}{1, x \text{ is rational}}\\{0, x \text{ is irrational}}\end{array}\right.$, is called the Dirichlet function. Now, a function similar to the Dirichlet function is defined as $L\left(x\right)=\left\{\begin{array}{l}{x, x \text{ is rational}}\\{0, x \text{ is irrational}}\end{array}\right.$. There are four conclusions about the Dirichlet function and the $L$ function:<br/>$(1)D\left(1\right)=L\left(1\right)$;<br/>$(2)$ The function $L\left(x\right)$ is an even function;<br/>$(3)$ There exist four points $A$, $B$, $C$, $D$ on the graph of the $L$ function such that the quadrilateral $ABCD$ is a rhombus;<br/>$(4)$ There exist three points $A$, $B$, $C$ on the graph of the $L$ function such that $\triangle ABC$ is an equilateral triangle.<br/>The correct numbers of the conclusions are ____.

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