
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
Definition: When $x$ takes any real number, and the function value is always not less than a constant, the function is called a "constant function," and this constant is called the "constant value."  
$(1)$ Determine: whether the function $y=x^{2}+2x+2$ is a "constant function." If it is, find the "constant value"; if not, please explain the reason.  
$(2)$ Given a "constant function" $y=3|ax^{2}+bx+c|+2$.  
① When $a \gt 0$ and $c \lt 0$, the "constant value" is ______;  
② If the sum of three integers $a$, $b$, $c$ is $12$, and $\frac{b}{a}=\frac{c}{b}$, find the maximum and minimum values of $a$, and find the corresponding values of $b$ and $c.  
$(3)$ For the "constant function" $y=ax^{2}+bx+c\left(b \gt a\right)$ with a "constant value" of $0$, and $\frac{{a+b+c}}{{a+b}}＞m$ always holds, find the range of values for $m$.

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