
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 $C(A)$ represent the number of elements in the non-empty set $A$. Define $A*B=\left\{\begin{array}{l}C(A)-C(B),C(A)≥C(B)\\ C(B)-C(A),C(A)<C(B)\end{array}\right.$, if $A=\{1,2\}$, $B=\{x|\left(x+a\right)(x^{3}+ax^{2}+2x)=0\}$, and $A*B=1$, find the cardinality of the set $S$ of all possible values of the real number $a$.

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