
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 points $A$ and $B$ on the number line representing rational numbers $a$ and $b$ respectively, then $|a-b|$ represents the distance between points $A$ and $B$, for example: $|1-2|$ represents the distance between points $1$ and $2$ on the number line, so the distance between points $1$ and $2$ on the number line is $|1-2|=1$.  
Applications:  
$(1)$ $|2-5|=\_\_\_\_\_\_$; $|\left(-2\right)-5|=\_\_\_\_\_\_$;  
$(2)$ ① The geometric interpretation of the expression $|x-3|$ is the distance between the point representing the rational number $x$ on the number line and the point representing the rational number ______;  
② Find the values of $x$ that satisfy the equation $|x-1|=3$.  
$(3)$ ① $|x-1|+|x+2|=3$ represents that the sum of the distances of the rational number $x$ on the number line to the points corresponding to $1$ and $-2$ is $3$, then directly write down all integers $x$ that satisfy the condition;  
② Write down directly the minimum value of $|x+8|+|x-3|+|x-6|$.

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