
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 $A=\{x | (x-2)(x-a)=0, a\in \mathbb{R}\}$, $B=\{x | x(x-1)=0\}$.  
$(Ⅰ)$ If $a=1$, find $A\cap B$ and $A\cup B$;  
$(Ⅱ)$ Let $C=A\cup B$, if the set $C$ has $8$ subsets, find the set of possible values for $a$.

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