
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 $A=\{1,2,3\}$, $B=\{x|ax-1\geqslant 0\}$, where $a\in R$.  
$(1)$ When $a=3$, find $A\cap B$ and $A\cup B$;  
$(2)$ If ___, find the range of real number $a$.  
Please choose one of the following conditions to fill in the blank in (2) and complete the solution:  
① $A\cap B=A$;  
② For all $x\in A$, $x\notin B$;  
③ "$x\in B$" is a necessary condition for "$x\in A$".

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