
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 functions $f\left(x\right)=x^{2}+ax+b(a$,$b\in R)$ and $g\left(x\right)=2^{x}-2^{2-x}+1$.<br/>$(1)$ Let the solution set of $f\left(x\right)\leqslant 0$ be $A$, and the solution set of $f\left[f\left(x\right)\right]\leqslant 3$ be $B$. If $A=B\neq \varnothing$, find the range of real number $a$;<br/>$(2)$ Given that the graph of function $h\left(x\right)$ is symmetric about the point $\left(1,1\right)$, and when $x\in \left[0,1\right]$, $h\left(x\right)=f\left(x\right)$. If for any $x_{1}\in \left[0,2\right]$, there always exists $x_{2}\in \left[0,2\right]$ such that $h(x_{1})=g(x_{2})$, find the range of real number $b$.

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