
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
【Background】The number line is an important tool in middle school mathematics. By using the number line, numbers and shapes can be perfectly combined. When studying the number line, we have discovered many important rules: If points $A$ and $B$ on the number line represent numbers $a$ and $b$ respectively, then the distance between points $A$ and $B$, denoted as $AB$, is $|a-b|$. If $a > b$, it can be simplified to $AB = a-b$. <br/>$(1)$ Given points $A$ and $B$ on the number line representing numbers $-10$ and $8$ respectively, the distance between points $A$ and $B$ is ______; <br/>【Exploration】To find the minimum value of the algebraic expression $|x+1|+|x-3|$, we can consider $|x+1|$ as the distance between numbers $x$ and $-1$ on the number line, and $|x-3|$ as the distance between numbers $x$ and $3$ on the number line, and discuss the following: <br/>When $x$ is between $-1$ and $3$, $|x+1|+|x-3|=4$; <br/>When $x$ is to the left of $-1$, $|x+1|+|x-3 \gt 4$; <br/>When $x$ is to the right of $3$, $|x+1|+|x-3 \gt 4$; <br/>In conclusion, the minimum value of the algebraic expression is $4$; <br/>$(2)$ The minimum value of $|x-3|+|x+2|$ is ______; <br/>【Application of Methods】: <br/>$(3)$ Given $|y-3|+|y+1|=8$, then $y=$______; <br/>【Transfer Application】: <br/>$(4)$ If $m$ and $n$ are integers, and $m$ and $n$ satisfy $\left(|m+1|+|2-m|\right)\left(|n-1|+|n+3|\right)=12$, then when $m=$______, $n=$______, the maximum value of $mn$ is ______.

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