
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
The parabola \( C_{1}: y = x^{2} + 2ax + b \) intersects the x-axis at points \( A \) and \( B \). If the vertex of \( C_{1} \) always lies within the circle \( C_{2} \) which has the line segment \( AB \) as its diameter, find the relationship that \( a \) and \( b \) must satisfy.

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