
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 $A=\{1,2\}$, $B=\{x|(x^{2}+ax)\cdot (x^{2}+ax+2)=0\}$, and $A*B=1$. Determine the set of all possible values of the real number $a$ denoted as $S$, and then find the number of elements in $S$, $C(S)$.

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