
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 number line is a very important mathematical tool, which establishes a correspondence between numbers and points on the number line, revealing the inherent connection between numbers and points. It is the foundation of the integration of "numbers and shapes." For example, from the "shape" perspective: $|3-1|$ can be understood as the distance between the two points representing $3$ and $1$ on the number line; $|3+1|$ can be understood as the distance between the two points representing $3$ and $-1$ on the number line. From the "number" perspective: the distance between the points representing $4$ and $-3$ on the number line can be expressed algebraically as $4-\left(-3\right)$. Based on the reading material above, explore the following questions:  
$(1)$ The distance between the points representing $2$ and $-5$ on the number line is ______; $|-4+6|=\_\_\_\_\_\_$; $|-2-4|=\_\_\_\_\_\_$; (Write down the final results directly)  
$(2)$ The geometric meaning of the equation $|x+2|=4$ is ______, then the value of $x$ is ______;  
$(3)$ If $x$ represents a number on the number line, does the expression $|x+1|+|x-3|$ have a minimum value? If yes, find the minimum value; if not, state the reason.

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