
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
Let the domain of the function $y=lg({2-\frac{1}{{x+1}}})$ be set $A$, and the domain of the function $y=\sqrt{1-|x-a|}$ be set $B$.  
$(1)$ If the universal set is $R$, find the closure of $A$;  
$(2)$ If $A\cup B=R$, find the range of real number $a$.

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