
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 $A$, and define $A*B= \begin{cases} 
C(A)-C(B), & \text{when } C(A) \geq C(B) \\
C(B)-C(A), & \text{when } C(A) < C(B)
\end{cases}$. If $A=\{x|x^2-ax-1=0, a\in \mathbb{R}\}$ and $B=\{x||x^2+bx+1|=1, b\in \mathbb{R}\}$, let $S=\{b|A*B=1\}$, find $C(S)$.

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