
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
First read and then explore the related questions: $|5-2|$ represents the absolute value of the difference between $5$ and $2$, which can also be understood as the distance between the two points corresponding to $5$ and $2$ on the number line; $|5+2|$ (can be seen as $|5-(-2)|$) represents the absolute value of the difference between $5$ and $-2$, which can also be understood as the distance between the two points corresponding to $5$ and $-2$ on the number line. Therefore, when points $A$ and $B$ represent rational numbers $a$ and $b$ on the number line, the distance between points $A$ and $B$ is represented as $AB$, and the distance between points $A$ and $B$ on the number line is $AB=|a-b|$.

$(1)$ The distance between the two points $A$ and $B$ representing $-3$ and $4$ on the number line is ______; if $AB=3$, and the number represented by $A$ is $2$, then the number represented by $B$ is ______;

$(2)$ If the rational number represented by point $A$ is $x$, then when $x$ is ______, the values of $|x+4|$ and $|x-2|$ are equal;

$(3)$ When $x=$ ______, $|x-1|+|x-2|+|x-3|+\ldots +|x-2023|$ has the minimum value (please directly write the value of $x$);

$(4)$ Find the minimum value in (3).

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