
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 $A=\{x|x^2-ax+a^2-19=0\}$, $B=\{x|x^2-5x+6=0\}$, and $C=\{x|x^2+2x-8=0\}$.

(1) If $A\cap B=A\cup B$, find the value of $a$;  
(2) If $\emptyset \subset (A\cap B)$ and $A\cap C=\emptyset$, find the value of $a$;  
(3) If $A\cap B=A\cap C\neq \emptyset$, find the value of $a$.

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