
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
Given a quadratic function $f\left(x\right)=ax^{2}+bx+c\left(a\neq 0\right)$, it is known that $f\left(x+1\right)-f\left(x\right)=2x+2$, and $f\left(0\right)=-2$. Find:<br/>
$(1)$ The analytical expression of the function $f\left(x\right)$;<br/>
$(2)$ Let $g\left(x\right)=f\left(x\right)-mx$. If the maximum value of the function $g\left(x\right)$ on the interval $\left[1,2\right]$ is $3$, find the value of the real number $m$;<br/>
$(3)$ Let $h\left(x\right)=[f\left(x\right)-x^{2}+2]\cdot |x-a|$, where $a\in R$. If the function $h\left(x\right)$ is monotonic on $\left[-2,2\right]$, find the range of values for $a$.

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