
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
If points $A$ and $B$ on the number line represent rational numbers $a$ and $b$ respectively, then the distance between points $A$ and $B$ is represented by $AB$, which is $AB=|a-b|$.
Answer the following questions using the number line:
$(1)$ The distance between the points representing $2$ and $5$ on the number line is ______;
$(2)$ The distance between the points representing $x$ and $-6$ on the number line is ______;
$(3)$ If $x$ represents a rational number, and $-2 \lt x \lt 2$, then $|x-2|+|x+2|=$______;
$(4)$ If $x$ represents a rational number, and $|x-1|+|x+3| \gt 4$, then the range of rational number $x$ is ______;
$(5)$ If $x$ represents a rational number, then the minimum value of $|x-3|+|x+2|+|x+1|$ is ______, and at this point $x=$______;
$(6)$ When $|x-1|+|x+2|=10-|y-3|-|y+4|$, then the maximum value of $y$ is ______.

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