
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
Choose any one of the three conditions, and fill in the blank in question (2) with the selected condition to solve the following problem.  
Given sets $A=\{x| (x-a+1)(x-a-1)\leqslant 0\}$ and $B=\{x| |x-\frac{1}{2}|\leq\frac{3}{2}\}$;  
$(1)$ When $a=2$, find $A\cup B$;  
$(2)$ If ____, find the range of real numbers $a$. [Note: If multiple conditions are selected and answered separately, the first answer will be scored.]

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