
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 $C(A)$ denote the number of elements in a non-empty set $A$. Define
$$
A*B= \begin{cases} C(A)-C(B), & \text{if } C(A) \geq C(B) \\ C(B)-C(A), & \text{if } C(A) < C(B) \end{cases}
$$
If $A=\{1,2\}$, $B=\{x|(x^2+ax)(x^2+ax+2)=0\}$, and $A*B=1$, determine the set of all possible values of the real number $a$.

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