
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 sets $A=\{x|a\leqslant x\leqslant a+2\}$ and $B=\{x|x^{2}+x-2 \lt 0\}$.  
$(1)$ If $a=0$, find $A$⋂$(\complement _{R}B)$;  
$(2)$ If the proposition $P$: "$\forall x\in A$, $x\notin B$" is a true proposition, find the range of real number $a$.

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