
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 the function $f\left(x\right)=x^{2}+ax+2$, where $a\in \mathbb{R}$.  
$(1)$ If the solution set of the inequality $f\left(x\right)\leqslant 0$ is $\left[1,2\right]$, find the solution set of the inequality $f\left(x\right)\geqslant 1-x^{2}$.  
$(2)$ If for any $x\in \left[-1,1\right]$, the inequality $f\left(x\right)\leqslant 2a\left(x-1\right)+4$ always holds, find the range of real numbers for $a$.  
$(3)$ Given $g\left(x\right)=-x+m$, when $a=-3$, if for any $x_{1}\in \left[1,4\right]$, there always exists $x_{2}\in \left(1,8\right)$ such that $f(x_{1})=g(x_{2})$, find the range of real numbers for $m$.

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