
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 $f(x)=x^{2}+ax+2$, where $a\in \mathbb{R}$.  
$(1)$ If the solution set of the inequality $f(x)\leqslant 0$ is $\left[1,2\right]$, find the solution set of the inequality $f(x)\geqslant 1-x^{2}$.  
$(2)$ If the inequality $f(x)\leqslant 2a(x-1)+4$ holds for any $x\in \left[-1,1\right]$, find the range of real number $a$.

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