
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 rectangular coordinate system on a plane, let the graph of the quadratic function $f(x)=x^{2}+2x+b (x \in \mathbb{R})$ intersect with both coordinate axes at three points. Denote the circle that passes through these three points as $C$. Find:

1. The range of the real number $b$.
2. The equation of circle $C$.

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