
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
For rational numbers $x$, $y$, $a$, $t$, if $|x-a|+|y-a|=t$, then $t$ is called the "beautiful association number" of $x$ and $y$ with respect to $a$. For example, $|2-1|+|3-1|=3$, then the "beautiful association number" of $2$ and $3$ with respect to $1$ is $3$. <br/> $(1)$ The "beautiful association number" of $-3$ and $5$ with respect to $2$ is ______; <br/> $(2)$ If the "beautiful association number" of $x$ and $2$ with respect to $3$ is $4$, find the value of $x$; <br/> $(3)$ If the "beautiful association number" of $x_{0}$ and $x_{1}$ with respect to $1$ is $1$, the "beautiful association number" of $x_{1}$ and $x_{2}$ with respect to $2$ is $1$, the "beautiful association number" of $x_{2}$ and $x_{3}$ with respect to $3$ is $1$, the "beautiful association number" of $x_{3}$ and $x_{4}$ with respect to $4$ is $1$, the "beautiful association number" of $x_{4}$ and $x_{5}$ with respect to $5$ is $1$, then the minimum value of $x_{1}+x_{2}+x_{3}+x_{4}$ is ______.

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