
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
Students, we all know that $|5-2|$ 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; $|5+2|$ 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. Let's explore:

$(1)$ $|-4+6|=$______; $|-2-4|=$______;

$(2)$ Find all integers $x$ that satisfy $|x+2|+|x-1|=3$;

$(3)$ If the point representing the number $a$ on the number line is located between -4 and 6, find the value of $|a+4|+|a-6|$;

$(4)$ When $a=$______, the value of $|a-1|+|a+5|+|a-4|$ is minimized, and the minimum value is ______.

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