
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 all know that $|7-\left(-1\right)|$ represents the absolute value of the difference between $7$ and $-1$, which can also be understood as the distance between the two points representing $7$ and $-1$ on the number line. For example, the geometric meaning of $|x-6|$ is the distance between the point representing the rational number $x$ and the point representing the rational number $6$ on the number line. Explore the following:<br/>$(1)$ Find $|3-\left(-2\right)|=$______; if $|x+2|=3$, then $x=$______;<br/>$(2)$ The minimum value of $|x-1|+|x+3|$ is ________;<br/>$(3)$ When $x=$______, the minimum value of $|x+1|+|x-2|+|x-4|$ is ________;<br/>$(4)$ Given $\left(|x+1|+|x-2|\right)\times \left(|y-2|+|y+1|\right)\times \left(|z-3|+|z+1|\right)=36$, find the maximum and minimum values of $x+y+z$.

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