
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
An ellipse defined by the equation \( x^{2} + 2 y^{2} = 2 \) intersects a line given by \( x + 2 y - 1 = 0 \) at points \( B \) and \( C \). Given point \( A \) with coordinates \( (2,2) \), find the equation of the circle passing through points \( A \), \( B \), and \( C \).

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