
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 $|A|$ represent the number of elements in a non-empty set $A$. Define $A*B=\left\{{\begin{array}{l}{|A|-|B|,|A|\geq|B|}\\{|B|-|A|,|B|>|A|}\end{array}}\right.$. If $A=\{0,1\}$, $B=\{x|(x^{2}+ax)(x^{2}+ax+3)=0\}$, and $A*B=1$, then the set $S$ of all possible values of the real number $a$ is $S=$____ (please list the elements).

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