
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
In the Cartesian coordinate plane $(xOy)$, let $f(x)=-x^{2}-2x+b (x\in R)$ be a quadratic function whose graph intersects both coordinate axes at three points. The circle passing through these three points is denoted as $C$.

(1) Find the range of possible values for the real number $b$.
(2) Find the equation of circle $C$.

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