
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
Read the following material: We know that the geometric meaning of $a$ is the distance between the point corresponding to the number $a$ on the number line and the origin, and the distance between the point corresponding to the number $a$ and the point corresponding to the number $0$ on the number line. The conclusion $|a|=|a-0|$ can be extended to $|a-b|$, which represents the distance between the points corresponding to the numbers $a$ and $b$ on the number line. For example, given $|a-1|=2$, find the value of $a$.  
Solution: The point on the number line that is at a distance of $2$ from $1$ corresponds to $3$ and $-1$, so the value of $a$ is $3$ and $-1$.  
Following the solution in the reading material, solve the following problems:  
$(1)$ Given $|a+2|=4$, find the value of $a$;  
$(2)$ If the point representing $a$ on the number line is between $-4$ and $2$, what is the value of $|a+4|+|a-2|$?  
$(3)$ Under what conditions does $|a-1|+|a+2|$ have a minimum value, and what is the minimum value?

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