
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
If any algebraic expression, when the maximum value obtained within a given range is exactly within that range, then the algebraic expression is called a "harmonious algebraic expression" for that range. For example, for the algebraic expression $x^{2}$ with $-1\leqslant x\leqslant 1$, the expression $x^{2}$ has a maximum value of $1$ at $x=\pm 1$ and a minimum value of $0$ at $x=0$. In this case, both the maximum value of $1$ and the minimum value of $0$ are within the range $-1\leqslant x\leqslant 1$ (including endpoints), so the algebraic expression $x^{2}$ is a "harmonious algebraic expression" for $-1\leqslant x\leqslant 1$.

$(1)$ If the algebraic expression about $x$ is $|x-1|$, when $-2\leqslant x\leqslant 2$, the maximum value is ______; the minimum value is ______; the algebraic expression $|x-1|$ is ______ (fill in "is" or "is not") a "harmonious algebraic expression" for $-2\leqslant x\leqslant 2$.

$(2)$ The following algebraic expressions about $x$ are "harmonious algebraic expressions" for $-2\leqslant x\leqslant 2": ______;

① $-x+1$; ② $-x^{2}+2$; ③ $x^{2}+|x|-4$;

$(3)$ If the algebraic expression about $x$ is $\frac{a}{{|x|+1}}-2$ and it is a "harmonious algebraic expression" for $-2\leqslant x\leqslant 2", find the maximum and minimum values of $a$.

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