
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 $A=\{x|x^{2}-4x=0\}$ and $B=\{x|ax^{2}-2x+8=0\}$.  
$(1)$ Does there exist a real number $a$ such that $A\cup B=\{0,2,4\}$? If yes, find the value of $a$; if not, explain why.  
$(2)$ If $A\cap B=B$, find the range of real numbers for $a$.

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